| L(s) = 1 | − 5·3-s + 3·5-s − 9·7-s + 3·9-s − 80·11-s + 26·13-s − 15·15-s + 19·17-s + 84·19-s + 45·21-s + 196·23-s − 239·25-s − 40·27-s + 44·29-s − 86·31-s + 400·33-s − 27·35-s − 209·37-s − 130·39-s − 230·41-s − 287·43-s + 9·45-s + 435·47-s − 111·49-s − 95·51-s + 118·53-s − 240·55-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 0.268·5-s − 0.485·7-s + 1/9·9-s − 2.19·11-s + 0.554·13-s − 0.258·15-s + 0.271·17-s + 1.01·19-s + 0.467·21-s + 1.77·23-s − 1.91·25-s − 0.285·27-s + 0.281·29-s − 0.498·31-s + 2.11·33-s − 0.130·35-s − 0.928·37-s − 0.533·39-s − 0.876·41-s − 1.01·43-s + 0.0298·45-s + 1.35·47-s − 0.323·49-s − 0.260·51-s + 0.305·53-s − 0.588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 692224 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 5 T + 22 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 248 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 9 T + 192 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 80 T + 3650 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 19 T + 8688 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 84 T + 11130 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 196 T + 33326 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 44 T + 10094 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 86 T + 56518 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 209 T + 112120 T^{2} + 209 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 230 T + 149010 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 287 T + 92698 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 435 T + 192728 T^{2} - 435 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 118 T + 297410 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 368 T + 379266 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1058 T + 580378 T^{2} - 1058 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 68 T + 373930 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 131 T + 493328 T^{2} + 131 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 456 T + 542718 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1008 T + 1233294 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1958 T + 1961238 T^{2} + 1958 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 720 T + 899726 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 928 T + 943870 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.813278967619206055679723237484, −9.319499284172354490094845639702, −8.690575745351321752582709298952, −8.331002260396050721010931764949, −7.935250529032507360428747065525, −7.27826445201795444998474985548, −7.11145528861906989029965312862, −6.58508725825674101626681817323, −5.89484740307466743700789038238, −5.50580407107914811657684156438, −5.23980109933732563050668266044, −5.16052919469470520776141001360, −4.06525882184495625362183936423, −3.70945114684107574543167233522, −2.81311815052099493056604838131, −2.74366042504932949051645542020, −1.74628020280605270243765824030, −1.05238160253304729636769076051, 0, 0,
1.05238160253304729636769076051, 1.74628020280605270243765824030, 2.74366042504932949051645542020, 2.81311815052099493056604838131, 3.70945114684107574543167233522, 4.06525882184495625362183936423, 5.16052919469470520776141001360, 5.23980109933732563050668266044, 5.50580407107914811657684156438, 5.89484740307466743700789038238, 6.58508725825674101626681817323, 7.11145528861906989029965312862, 7.27826445201795444998474985548, 7.935250529032507360428747065525, 8.331002260396050721010931764949, 8.690575745351321752582709298952, 9.319499284172354490094845639702, 9.813278967619206055679723237484
