| L(s) = 1 | + 2-s + 4-s − 5-s − 3·7-s + 8-s − 10-s − 2·11-s − 5·13-s − 3·14-s + 16-s + 5·17-s − 20-s − 2·22-s + 3·23-s + 25-s − 5·26-s − 3·28-s − 7·29-s + 2·31-s + 32-s + 5·34-s + 3·35-s + 10·37-s − 40-s + 10·43-s − 2·44-s + 3·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 1.38·13-s − 0.801·14-s + 1/4·16-s + 1.21·17-s − 0.223·20-s − 0.426·22-s + 0.625·23-s + 1/5·25-s − 0.980·26-s − 0.566·28-s − 1.29·29-s + 0.359·31-s + 0.176·32-s + 0.857·34-s + 0.507·35-s + 1.64·37-s − 0.158·40-s + 1.52·43-s − 0.301·44-s + 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.20353099274415, −14.74797457669959, −14.38061885217476, −13.67135690402657, −12.98764153711838, −12.73075335177499, −12.35666095065343, −11.64980473854544, −11.24808246573514, −10.38685936501152, −10.10279764621698, −9.405736483423848, −9.033314650774247, −7.904324545743640, −7.545709125257126, −7.242805119753324, −6.320330091447912, −5.908796603884848, −5.226061214317723, −4.653294856505688, −3.992618461074705, −3.199542192945661, −2.874729612062208, −2.144090389562155, −0.9388154144038221, 0,
0.9388154144038221, 2.144090389562155, 2.874729612062208, 3.199542192945661, 3.992618461074705, 4.653294856505688, 5.226061214317723, 5.908796603884848, 6.320330091447912, 7.242805119753324, 7.545709125257126, 7.904324545743640, 9.033314650774247, 9.405736483423848, 10.10279764621698, 10.38685936501152, 11.24808246573514, 11.64980473854544, 12.35666095065343, 12.73075335177499, 12.98764153711838, 13.67135690402657, 14.38061885217476, 14.74797457669959, 15.20353099274415
