| L(s) = 1 | − 4-s − 9-s − 6·11-s + 16-s + 14·19-s − 12·29-s − 14·31-s + 36-s − 12·41-s + 6·44-s + 13·49-s − 20·61-s − 64-s − 12·71-s − 14·76-s − 34·79-s + 81-s + 24·89-s + 6·99-s − 30·101-s + 14·109-s + 12·116-s + 5·121-s + 14·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 1.80·11-s + 1/4·16-s + 3.21·19-s − 2.22·29-s − 2.51·31-s + 1/6·36-s − 1.87·41-s + 0.904·44-s + 13/7·49-s − 2.56·61-s − 1/8·64-s − 1.42·71-s − 1.60·76-s − 3.82·79-s + 1/9·81-s + 2.54·89-s + 0.603·99-s − 2.98·101-s + 1.34·109-s + 1.11·116-s + 5/11·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2686113729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2686113729\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.279353336671326402025373360324, −8.756441586066739600777039885811, −8.395875628819618426841910050780, −7.69990400716311455162435486876, −7.52431935486464251488106197899, −7.34587201565838584997364620744, −7.18922747516542450082361005697, −6.21514010011937852838357450134, −5.80541278797801446915087566900, −5.34506079162002874048220604551, −5.32972968541709628887106147473, −5.07838740298404740682706793087, −4.26653256048889989246882328442, −3.82056319352285405482983521168, −3.22297936132339876477210583767, −3.13277066085658569817202219749, −2.51957767955154180736522142661, −1.72632459164441281525344687421, −1.33781476584183833567644085153, −0.17387535493892419288918454133,
0.17387535493892419288918454133, 1.33781476584183833567644085153, 1.72632459164441281525344687421, 2.51957767955154180736522142661, 3.13277066085658569817202219749, 3.22297936132339876477210583767, 3.82056319352285405482983521168, 4.26653256048889989246882328442, 5.07838740298404740682706793087, 5.32972968541709628887106147473, 5.34506079162002874048220604551, 5.80541278797801446915087566900, 6.21514010011937852838357450134, 7.18922747516542450082361005697, 7.34587201565838584997364620744, 7.52431935486464251488106197899, 7.69990400716311455162435486876, 8.395875628819618426841910050780, 8.756441586066739600777039885811, 9.279353336671326402025373360324
