| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 11-s + 13-s + 16-s − 2·17-s + 4·19-s − 20-s + 22-s + 25-s − 26-s + 2·29-s + 8·31-s − 32-s + 2·34-s − 10·37-s − 4·38-s + 40-s − 10·41-s − 4·43-s − 44-s + 8·47-s − 7·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.223·20-s + 0.213·22-s + 1/5·25-s − 0.196·26-s + 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 1.64·37-s − 0.648·38-s + 0.158·40-s − 1.56·41-s − 0.609·43-s − 0.150·44-s + 1.16·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.74229335869920, −15.81277665806021, −15.55523334795625, −15.24707508920978, −14.22197064141300, −13.78702262793928, −13.22857229047201, −12.31949168752169, −11.97794423960940, −11.39355805910268, −10.76268367760842, −10.17738386663923, −9.720218018726515, −8.846922974345836, −8.448105005382635, −7.904935941242821, −7.087360399661944, −6.750192191296478, −5.883133996831976, −5.135821017753946, −4.454228860758402, −3.479896124399778, −2.935553737031577, −1.951680303855680, −1.050851811671839, 0,
1.050851811671839, 1.951680303855680, 2.935553737031577, 3.479896124399778, 4.454228860758402, 5.135821017753946, 5.883133996831976, 6.750192191296478, 7.087360399661944, 7.904935941242821, 8.448105005382635, 8.846922974345836, 9.720218018726515, 10.17738386663923, 10.76268367760842, 11.39355805910268, 11.97794423960940, 12.31949168752169, 13.22857229047201, 13.78702262793928, 14.22197064141300, 15.24707508920978, 15.55523334795625, 15.81277665806021, 16.74229335869920
