| L(s) = 1 | + 2-s − 4-s + 5-s + 4·7-s − 3·8-s + 10-s − 4·11-s − 6·13-s + 4·14-s − 16-s + 2·17-s − 20-s − 4·22-s + 23-s + 25-s − 6·26-s − 4·28-s − 10·29-s + 8·31-s + 5·32-s + 2·34-s + 4·35-s − 2·37-s − 3·40-s + 2·41-s − 8·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.51·7-s − 1.06·8-s + 0.316·10-s − 1.20·11-s − 1.66·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.223·20-s − 0.852·22-s + 0.208·23-s + 1/5·25-s − 1.17·26-s − 0.755·28-s − 1.85·29-s + 1.43·31-s + 0.883·32-s + 0.342·34-s + 0.676·35-s − 0.328·37-s − 0.474·40-s + 0.312·41-s − 1.21·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 373635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.81585299485167, −12.31802017494972, −11.86957920266510, −11.51110147888113, −11.01049338377470, −10.33274262471199, −10.12153426650246, −9.539602655426242, −9.184680843529376, −8.524786115802803, −8.039117692591735, −7.818084035244027, −7.253680550409273, −6.739279792323628, −5.944757272623878, −5.471891378976946, −5.225474348006083, −4.762788525355892, −4.507889061140586, −3.827775104582738, −3.045736925884105, −2.726870631335246, −2.051549679868420, −1.622939670425181, −0.6880826530301070, 0,
0.6880826530301070, 1.622939670425181, 2.051549679868420, 2.726870631335246, 3.045736925884105, 3.827775104582738, 4.507889061140586, 4.762788525355892, 5.225474348006083, 5.471891378976946, 5.944757272623878, 6.739279792323628, 7.253680550409273, 7.818084035244027, 8.039117692591735, 8.524786115802803, 9.184680843529376, 9.539602655426242, 10.12153426650246, 10.33274262471199, 11.01049338377470, 11.51110147888113, 11.86957920266510, 12.31802017494972, 12.81585299485167
