| L(s) = 1 | + 2.29·2-s + 3.28·4-s + 5-s − 2.76·7-s + 2.94·8-s + 2.29·10-s + 1.67·11-s − 6.25·13-s − 6.35·14-s + 0.200·16-s − 4.10·17-s − 4.67·19-s + 3.28·20-s + 3.83·22-s − 9.46·23-s + 25-s − 14.3·26-s − 9.06·28-s + 4.18·29-s + 3.14·31-s − 5.42·32-s − 9.43·34-s − 2.76·35-s + 10.3·37-s − 10.7·38-s + 2.94·40-s − 6.63·41-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.64·4-s + 0.447·5-s − 1.04·7-s + 1.04·8-s + 0.726·10-s + 0.503·11-s − 1.73·13-s − 1.69·14-s + 0.0501·16-s − 0.995·17-s − 1.07·19-s + 0.733·20-s + 0.818·22-s − 1.97·23-s + 0.200·25-s − 2.81·26-s − 1.71·28-s + 0.777·29-s + 0.565·31-s − 0.958·32-s − 1.61·34-s − 0.467·35-s + 1.70·37-s − 1.74·38-s + 0.465·40-s − 1.03·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 2.29T + 2T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 6.25T + 13T^{2} \) |
| 17 | \( 1 + 4.10T + 17T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 + 9.46T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 - 3.14T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 6.63T + 41T^{2} \) |
| 43 | \( 1 - 4.38T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 - 2.86T + 53T^{2} \) |
| 59 | \( 1 + 1.93T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 0.329T + 73T^{2} \) |
| 79 | \( 1 - 6.42T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.85464072669415602258671024800, −6.86189621938999477045886800194, −6.36207719878628973484034007258, −5.98196816690998524441227342060, −4.83049766173098906954379030522, −4.42151998522278155805063510319, −3.56340869998740908701080264638, −2.56163416257665253102779254823, −2.12756476698281169440666134235, 0,
2.12756476698281169440666134235, 2.56163416257665253102779254823, 3.56340869998740908701080264638, 4.42151998522278155805063510319, 4.83049766173098906954379030522, 5.98196816690998524441227342060, 6.36207719878628973484034007258, 6.86189621938999477045886800194, 7.85464072669415602258671024800
