| L(s) = 1 | + 0.679·2-s + 0.178·3-s − 1.53·4-s + 5-s + 0.121·6-s + 1.84·7-s − 2.40·8-s − 2.96·9-s + 0.679·10-s − 0.320·11-s − 0.275·12-s + 0.954·13-s + 1.25·14-s + 0.178·15-s + 1.44·16-s − 5.78·17-s − 2.01·18-s + 6.14·19-s − 1.53·20-s + 0.330·21-s − 0.217·22-s − 0.429·24-s + 25-s + 0.648·26-s − 1.06·27-s − 2.84·28-s − 7.37·29-s + ⋯ |
| L(s) = 1 | + 0.480·2-s + 0.103·3-s − 0.769·4-s + 0.447·5-s + 0.0496·6-s + 0.697·7-s − 0.850·8-s − 0.989·9-s + 0.214·10-s − 0.0965·11-s − 0.0793·12-s + 0.264·13-s + 0.335·14-s + 0.0461·15-s + 0.360·16-s − 1.40·17-s − 0.475·18-s + 1.41·19-s − 0.343·20-s + 0.0720·21-s − 0.0464·22-s − 0.0877·24-s + 0.200·25-s + 0.127·26-s − 0.205·27-s − 0.536·28-s − 1.36·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.679T + 2T^{2} \) |
| 3 | \( 1 - 0.178T + 3T^{2} \) |
| 7 | \( 1 - 1.84T + 7T^{2} \) |
| 11 | \( 1 + 0.320T + 11T^{2} \) |
| 13 | \( 1 - 0.954T + 13T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 - 6.14T + 19T^{2} \) |
| 29 | \( 1 + 7.37T + 29T^{2} \) |
| 31 | \( 1 + 4.03T + 31T^{2} \) |
| 37 | \( 1 + 6.16T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 - 1.48T + 43T^{2} \) |
| 47 | \( 1 - 9.26T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 3.57T + 59T^{2} \) |
| 61 | \( 1 + 1.28T + 61T^{2} \) |
| 67 | \( 1 + 8.07T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 2.66T + 83T^{2} \) |
| 89 | \( 1 + 7.64T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.647961175959982767784537208523, −7.84220173482585896010610801983, −6.89656428557395528259269977549, −5.78364561279797780483892218219, −5.41432621591596608609811342149, −4.61212711165320839833457613075, −3.67000917380202721269223696867, −2.80893337186891956309651335218, −1.63231749224219710093304436141, 0,
1.63231749224219710093304436141, 2.80893337186891956309651335218, 3.67000917380202721269223696867, 4.61212711165320839833457613075, 5.41432621591596608609811342149, 5.78364561279797780483892218219, 6.89656428557395528259269977549, 7.84220173482585896010610801983, 8.647961175959982767784537208523
