| L(s) = 1 | + 2-s + 1.41·3-s + 4-s + 1.41·5-s + 1.41·6-s + 8-s − 0.999·9-s + 1.41·10-s − 2·11-s + 1.41·12-s − 4.24·13-s + 2.00·15-s + 16-s − 2.82·17-s − 0.999·18-s − 3.74·19-s + 1.41·20-s − 2·22-s − 7.29·23-s + 1.41·24-s − 2.99·25-s − 4.24·26-s − 5.65·27-s + 6.93·29-s + 2.00·30-s + 8.89·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.816·3-s + 0.5·4-s + 0.632·5-s + 0.577·6-s + 0.353·8-s − 0.333·9-s + 0.447·10-s − 0.603·11-s + 0.408·12-s − 1.17·13-s + 0.516·15-s + 0.250·16-s − 0.685·17-s − 0.235·18-s − 0.858·19-s + 0.316·20-s − 0.426·22-s − 1.52·23-s + 0.288·24-s − 0.599·25-s − 0.832·26-s − 1.08·27-s + 1.28·29-s + 0.365·30-s + 1.59·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 23 | \( 1 + 7.29T + 23T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 - 8.89T + 31T^{2} \) |
| 37 | \( 1 + 0.354T + 37T^{2} \) |
| 41 | \( 1 + 0.500T + 41T^{2} \) |
| 43 | \( 1 + 5.64T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 2.93T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 9.39T + 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 + 6.58T + 71T^{2} \) |
| 73 | \( 1 + 0.913T + 73T^{2} \) |
| 83 | \( 1 + 3.74T + 83T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 + 1.41T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66206504171725252791317663517, −6.63949020799732337843797873421, −6.12371617440249194703547732465, −5.40185974542953616923966006566, −4.57363630652954852334619686093, −4.02714913751285763626447043311, −2.79399711917467332032282901544, −2.55168075623193777540880155211, −1.78101169518305220235623992028, 0,
1.78101169518305220235623992028, 2.55168075623193777540880155211, 2.79399711917467332032282901544, 4.02714913751285763626447043311, 4.57363630652954852334619686093, 5.40185974542953616923966006566, 6.12371617440249194703547732465, 6.63949020799732337843797873421, 7.66206504171725252791317663517
