| L(s) = 1 | + 0.171·2-s − 1.97·4-s − 5-s + 3·7-s − 0.681·8-s − 0.171·10-s + 0.510·11-s − 0.882·13-s + 0.515·14-s + 3.82·16-s + 3.48·17-s − 6.67·19-s + 1.97·20-s + 0.0875·22-s − 4.11·23-s + 25-s − 0.151·26-s − 5.91·28-s − 9.10·29-s + 6.98·31-s + 2.02·32-s + 0.597·34-s − 3·35-s − 10.2·37-s − 1.14·38-s + 0.681·40-s + 2.20·41-s + ⋯ |
| L(s) = 1 | + 0.121·2-s − 0.985·4-s − 0.447·5-s + 1.13·7-s − 0.241·8-s − 0.0542·10-s + 0.153·11-s − 0.244·13-s + 0.137·14-s + 0.955·16-s + 0.844·17-s − 1.53·19-s + 0.440·20-s + 0.0186·22-s − 0.858·23-s + 0.200·25-s − 0.0297·26-s − 1.11·28-s − 1.69·29-s + 1.25·31-s + 0.357·32-s + 0.102·34-s − 0.507·35-s − 1.68·37-s − 0.185·38-s + 0.107·40-s + 0.344·41-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 | \( 1 - 0.171T + 2T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 0.510T + 11T^{2} \) |
| 13 | \( 1 + 0.882T + 13T^{2} \) |
| 17 | \( 1 - 3.48T + 17T^{2} \) |
| 19 | \( 1 + 6.67T + 19T^{2} \) |
| 23 | \( 1 + 4.11T + 23T^{2} \) |
| 29 | \( 1 + 9.10T + 29T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 - 8.21T + 47T^{2} \) |
| 53 | \( 1 - 1.35T + 53T^{2} \) |
| 59 | \( 1 - 7.92T + 59T^{2} \) |
| 61 | \( 1 - 3.09T + 61T^{2} \) |
| 67 | \( 1 + 5.64T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 6.45T + 73T^{2} \) |
| 79 | \( 1 + 4.74T + 79T^{2} \) |
| 83 | \( 1 + 4.60T + 83T^{2} \) |
| 97 | \( 1 - 4.91T + 97T^{2} \) |
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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
−8.111505792448270561782470484236, −7.63733253576898020254173609227, −6.60950061485368870570065030444, −5.57810512821987080588174735832, −5.09100217785884728874366176245, −4.10107667655840031973647414297, −3.85686199353954603742408054571, −2.44389128229036160168518411370, −1.33402326952032983160699866376, 0,
1.33402326952032983160699866376, 2.44389128229036160168518411370, 3.85686199353954603742408054571, 4.10107667655840031973647414297, 5.09100217785884728874366176245, 5.57810512821987080588174735832, 6.60950061485368870570065030444, 7.63733253576898020254173609227, 8.111505792448270561782470484236
