| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s + 4·10-s − 11-s + 13-s − 4·16-s − 17-s − 2·19-s − 4·20-s + 2·22-s − 4·23-s − 25-s − 2·26-s − 5·31-s + 8·32-s + 2·34-s + 5·37-s + 4·38-s + 9·41-s − 13·43-s − 2·44-s + 8·46-s − 6·47-s + 2·50-s + 2·52-s + 3·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s + 1.26·10-s − 0.301·11-s + 0.277·13-s − 16-s − 0.242·17-s − 0.458·19-s − 0.894·20-s + 0.426·22-s − 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.898·31-s + 1.41·32-s + 0.342·34-s + 0.821·37-s + 0.648·38-s + 1.40·41-s − 1.98·43-s − 0.301·44-s + 1.17·46-s − 0.875·47-s + 0.282·50-s + 0.277·52-s + 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.97281634652173, −13.56501845781600, −12.94184302457572, −12.50632933439204, −11.81172596657093, −11.44113168131928, −10.97042256068127, −10.57780959548652, −10.00073222504085, −9.525048915417879, −9.040945383599811, −8.505226478205663, −8.036629213470224, −7.671738154089454, −7.331267329152710, −6.429725543539203, −6.265959002500888, −5.299853667890346, −4.676194963875212, −4.109571556524308, −3.583094361628377, −2.796833436600531, −2.032672093995381, −1.530084828887845, −0.5778435724192416, 0,
0.5778435724192416, 1.530084828887845, 2.032672093995381, 2.796833436600531, 3.583094361628377, 4.109571556524308, 4.676194963875212, 5.299853667890346, 6.265959002500888, 6.429725543539203, 7.331267329152710, 7.671738154089454, 8.036629213470224, 8.505226478205663, 9.040945383599811, 9.525048915417879, 10.00073222504085, 10.57780959548652, 10.97042256068127, 11.44113168131928, 11.81172596657093, 12.50632933439204, 12.94184302457572, 13.56501845781600, 13.97281634652173
