L(s) = 1 | + 2-s + 4-s + 8-s + 4.24·11-s − 6.24·13-s + 16-s − 6.24·19-s + 4.24·22-s + 7.24·23-s − 5·25-s − 6.24·26-s − 4.24·29-s − 0.757·31-s + 32-s − 4·37-s − 6.24·38-s + 5.48·41-s − 6.48·43-s + 4.24·44-s + 7.24·46-s − 13.2·47-s − 5·50-s − 6.24·52-s + 4.24·53-s − 4.24·58-s − 6.24·61-s − 0.757·62-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.353·8-s + 1.27·11-s − 1.73·13-s + 0.250·16-s − 1.43·19-s + 0.904·22-s + 1.51·23-s − 25-s − 1.22·26-s − 0.787·29-s − 0.136·31-s + 0.176·32-s − 0.657·37-s − 1.01·38-s + 0.856·41-s − 0.988·43-s + 0.639·44-s + 1.06·46-s − 1.93·47-s − 0.707·50-s − 0.865·52-s + 0.582·53-s − 0.557·58-s − 0.799·61-s − 0.0961·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 0.757T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 5.48T + 41T^{2} \) |
| 43 | \( 1 + 6.48T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 + 7.24T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 + 7.75T + 83T^{2} \) |
| 89 | \( 1 - 5.48T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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
−7.29380536921522461791324028112, −6.74760429291660929004530531662, −6.17269066006716200603389953134, −5.24784604285028041449704434931, −4.67343780110178242937896718250, −3.99599631547843257542725389735, −3.20595472947644881187796041516, −2.28799448991214395916509928292, −1.53912586395349666517643384759, 0,
1.53912586395349666517643384759, 2.28799448991214395916509928292, 3.20595472947644881187796041516, 3.99599631547843257542725389735, 4.67343780110178242937896718250, 5.24784604285028041449704434931, 6.17269066006716200603389953134, 6.74760429291660929004530531662, 7.29380536921522461791324028112