| L(s) = 1 | − 2-s + 2.27·3-s + 4-s − 2.27·6-s + 1.27·7-s − 8-s + 2.16·9-s − 4.43·11-s + 2.27·12-s − 5.27·13-s − 1.27·14-s + 16-s − 0.273·17-s − 2.16·18-s − 5.71·19-s + 2.89·21-s + 4.43·22-s − 6.54·23-s − 2.27·24-s + 5.27·26-s − 1.89·27-s + 1.27·28-s + 0.546·29-s − 5.98·31-s − 32-s − 10.0·33-s + 0.273·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.31·3-s + 0.5·4-s − 0.927·6-s + 0.481·7-s − 0.353·8-s + 0.722·9-s − 1.33·11-s + 0.656·12-s − 1.46·13-s − 0.340·14-s + 0.250·16-s − 0.0662·17-s − 0.510·18-s − 1.31·19-s + 0.631·21-s + 0.946·22-s − 1.36·23-s − 0.463·24-s + 1.03·26-s − 0.364·27-s + 0.240·28-s + 0.101·29-s − 1.07·31-s − 0.176·32-s − 1.75·33-s + 0.0468·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 2.27T + 3T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 + 5.27T + 13T^{2} \) |
| 17 | \( 1 + 0.273T + 17T^{2} \) |
| 19 | \( 1 + 5.71T + 19T^{2} \) |
| 23 | \( 1 + 6.54T + 23T^{2} \) |
| 29 | \( 1 - 0.546T + 29T^{2} \) |
| 31 | \( 1 + 5.98T + 31T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 2.33T + 53T^{2} \) |
| 59 | \( 1 - 2.37T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 7.15T + 67T^{2} \) |
| 71 | \( 1 - 5.60T + 71T^{2} \) |
| 73 | \( 1 + 9.71T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 3.89T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 0.879T + 97T^{2} \) |
| 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
−8.734671166136427204870632918819, −7.994495575888687250059781649198, −7.77248253254193019822401590837, −6.83465208904565015238105270550, −5.62578311779360009185543949590, −4.66867185435946907554061808349, −3.56668292369494028811310697636, −2.31772836902885995639686810395, −2.17219856844586864990858923792, 0,
2.17219856844586864990858923792, 2.31772836902885995639686810395, 3.56668292369494028811310697636, 4.66867185435946907554061808349, 5.62578311779360009185543949590, 6.83465208904565015238105270550, 7.77248253254193019822401590837, 7.994495575888687250059781649198, 8.734671166136427204870632918819
