| L(s) = 1 | − 2.41·2-s + 3.84·4-s − 2.86·5-s − 7-s − 4.46·8-s + 6.92·10-s + 5.94·11-s + 5.81·13-s + 2.41·14-s + 3.10·16-s + 2.46·17-s + 4.96·19-s − 11.0·20-s − 14.3·22-s − 5.91·23-s + 3.21·25-s − 14.0·26-s − 3.84·28-s − 9.12·29-s + 1.52·31-s + 1.42·32-s − 5.96·34-s + 2.86·35-s + 10.9·37-s − 11.9·38-s + 12.7·40-s − 7.45·41-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.92·4-s − 1.28·5-s − 0.377·7-s − 1.57·8-s + 2.19·10-s + 1.79·11-s + 1.61·13-s + 0.646·14-s + 0.775·16-s + 0.598·17-s + 1.13·19-s − 2.46·20-s − 3.06·22-s − 1.23·23-s + 0.642·25-s − 2.75·26-s − 0.726·28-s − 1.69·29-s + 0.273·31-s + 0.252·32-s − 1.02·34-s + 0.484·35-s + 1.79·37-s − 1.94·38-s + 2.02·40-s − 1.16·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 2.86T + 5T^{2} \) |
| 11 | \( 1 - 5.94T + 11T^{2} \) |
| 13 | \( 1 - 5.81T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 + 5.91T + 23T^{2} \) |
| 29 | \( 1 + 9.12T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 7.45T + 41T^{2} \) |
| 43 | \( 1 + 6.37T + 43T^{2} \) |
| 47 | \( 1 - 3.77T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 2.34T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 9.61T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 2.71T + 83T^{2} \) |
| 89 | \( 1 - 1.78T + 89T^{2} \) |
| 97 | \( 1 + 5.89T + 97T^{2} \) |
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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
−7.58642643733980357293716027794, −7.19356943394381143985874158713, −6.27471092474965026563596280572, −5.91926170070473769034500855868, −4.35283992954374738017727562706, −3.69875446937873180175521976729, −3.16970378013248089355482635562, −1.63349567287706024824639619902, −1.11846487088027115600647743453, 0,
1.11846487088027115600647743453, 1.63349567287706024824639619902, 3.16970378013248089355482635562, 3.69875446937873180175521976729, 4.35283992954374738017727562706, 5.91926170070473769034500855868, 6.27471092474965026563596280572, 7.19356943394381143985874158713, 7.58642643733980357293716027794
