| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2.44·11-s + 1.44·13-s − 14-s + 16-s + 17-s − 4.44·19-s − 2.44·22-s + 1.44·23-s + 1.44·26-s − 28-s − 2.55·29-s − 1.44·31-s + 32-s + 34-s + 3.34·37-s − 4.44·38-s − 4.89·41-s − 3.89·43-s − 2.44·44-s + 1.44·46-s + 11.7·47-s + 49-s + 1.44·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.377·7-s + 0.353·8-s − 0.738·11-s + 0.402·13-s − 0.267·14-s + 0.250·16-s + 0.242·17-s − 1.02·19-s − 0.522·22-s + 0.302·23-s + 0.284·26-s − 0.188·28-s − 0.473·29-s − 0.260·31-s + 0.176·32-s + 0.171·34-s + 0.550·37-s − 0.721·38-s − 0.765·41-s − 0.594·43-s − 0.369·44-s + 0.213·46-s + 1.72·47-s + 0.142·49-s + 0.201·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 23 | \( 1 - 1.44T + 23T^{2} \) |
| 29 | \( 1 + 2.55T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 + 3.89T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 5.44T + 53T^{2} \) |
| 59 | \( 1 + 0.101T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 - 5.44T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 6.44T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 11T + 89T^{2} \) |
| 97 | \( 1 + 9.55T + 97T^{2} \) |
| 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
−7.13118123961316428919462994785, −6.70449515477272964278672457636, −5.77638699855281663981503800813, −5.46158304410155166259654822598, −4.47342034800037097658573544578, −3.91824062602671063903240934109, −3.04767942455355005141932825649, −2.40678780384455510814128033190, −1.39301760508774699725673714588, 0,
1.39301760508774699725673714588, 2.40678780384455510814128033190, 3.04767942455355005141932825649, 3.91824062602671063903240934109, 4.47342034800037097658573544578, 5.46158304410155166259654822598, 5.77638699855281663981503800813, 6.70449515477272964278672457636, 7.13118123961316428919462994785
