| L(s) = 1 | + 3-s + 2·5-s + 2·7-s − 4·9-s − 4·11-s + 2·13-s + 2·15-s − 7·17-s − 3·19-s + 2·21-s − 8·23-s + 3·25-s − 6·27-s − 7·29-s + 5·31-s − 4·33-s + 4·35-s + 3·37-s + 2·39-s − 15·41-s − 6·43-s − 8·45-s + 3·49-s − 7·51-s − 8·55-s − 3·57-s + 7·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s − 4/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.69·17-s − 0.688·19-s + 0.436·21-s − 1.66·23-s + 3/5·25-s − 1.15·27-s − 1.29·29-s + 0.898·31-s − 0.696·33-s + 0.676·35-s + 0.493·37-s + 0.320·39-s − 2.34·41-s − 0.914·43-s − 1.19·45-s + 3/7·49-s − 0.980·51-s − 1.07·55-s − 0.397·57-s + 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13249600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13249600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 39 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 69 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 45 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 15 T + 137 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 29 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 135 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 18 T + 178 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 13 T + 99 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + T + 117 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18 T + 230 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−8.307065685524497764506402086081, −8.289360370353709283163546443505, −7.67658273025221583112434395700, −7.42914088619605982098954106374, −6.76650221980385883362212150189, −6.45751529802274101618450950947, −5.99477111280608324298256454866, −5.86766888870298747359816389051, −5.27588964375858326568990371452, −5.10703623115082235947220807912, −4.36470102186110479470418010520, −4.34370917078153160969723114031, −3.53248685223367240810849843416, −3.16837791223600599574991657207, −2.54147264833389378701914769471, −2.37512246164586866441784118170, −1.81958380998421108718341058169, −1.51109463660980875131059846008, 0, 0,
1.51109463660980875131059846008, 1.81958380998421108718341058169, 2.37512246164586866441784118170, 2.54147264833389378701914769471, 3.16837791223600599574991657207, 3.53248685223367240810849843416, 4.34370917078153160969723114031, 4.36470102186110479470418010520, 5.10703623115082235947220807912, 5.27588964375858326568990371452, 5.86766888870298747359816389051, 5.99477111280608324298256454866, 6.45751529802274101618450950947, 6.76650221980385883362212150189, 7.42914088619605982098954106374, 7.67658273025221583112434395700, 8.289360370353709283163546443505, 8.307065685524497764506402086081
