| L(s) = 1 | + 5-s + 7-s − 2·11-s + 5·13-s + 8·17-s − 10·19-s − 2·23-s + 10·29-s + 8·31-s + 35-s − 6·37-s + 6·41-s − 4·43-s − 8·47-s + 7·49-s − 12·53-s − 2·55-s − 4·59-s + 5·61-s + 5·65-s + 7·67-s − 12·71-s − 18·73-s − 2·77-s − 3·79-s + 2·83-s + 8·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.603·11-s + 1.38·13-s + 1.94·17-s − 2.29·19-s − 0.417·23-s + 1.85·29-s + 1.43·31-s + 0.169·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 49-s − 1.64·53-s − 0.269·55-s − 0.520·59-s + 0.640·61-s + 0.620·65-s + 0.855·67-s − 1.42·71-s − 2.10·73-s − 0.227·77-s − 0.337·79-s + 0.219·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.110475953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.110475953\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−8.712607036135152781559145359729, −8.463515552407996208283127995058, −8.104743937011490115203757494307, −7.978997333339432818276948388429, −7.32339905040966103274605126905, −7.00815450400771221949493473984, −6.40870525303038711965989394035, −6.11779289366797645606845046099, −5.97175004030071896383000131913, −5.58488565043829121858857465160, −4.75244905254776834002228683580, −4.75179072627544156992910344013, −4.31601563699528076500694039246, −3.65866775998458643963262225566, −3.22740875260221110389281653390, −2.90634465987910073595815602894, −2.24408207472354851537028761442, −1.74251376984095263425476971971, −1.24847036278373727667627163957, −0.57467501467699531115302204025,
0.57467501467699531115302204025, 1.24847036278373727667627163957, 1.74251376984095263425476971971, 2.24408207472354851537028761442, 2.90634465987910073595815602894, 3.22740875260221110389281653390, 3.65866775998458643963262225566, 4.31601563699528076500694039246, 4.75179072627544156992910344013, 4.75244905254776834002228683580, 5.58488565043829121858857465160, 5.97175004030071896383000131913, 6.11779289366797645606845046099, 6.40870525303038711965989394035, 7.00815450400771221949493473984, 7.32339905040966103274605126905, 7.978997333339432818276948388429, 8.104743937011490115203757494307, 8.463515552407996208283127995058, 8.712607036135152781559145359729
