| L(s) = 1 | − 2-s + 4-s + 1.87·5-s − 3.28·7-s − 8-s − 1.87·10-s + 4.43·13-s + 3.28·14-s + 16-s − 6.72·17-s + 1.69·19-s + 1.87·20-s + 1.69·23-s − 1.47·25-s − 4.43·26-s − 3.28·28-s − 9.75·29-s + 1.59·31-s − 32-s + 6.72·34-s − 6.16·35-s + 4.53·37-s − 1.69·38-s − 1.87·40-s − 3.33·41-s + 7.16·43-s − 1.69·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.839·5-s − 1.24·7-s − 0.353·8-s − 0.593·10-s + 1.23·13-s + 0.877·14-s + 0.250·16-s − 1.63·17-s + 0.388·19-s + 0.419·20-s + 0.353·23-s − 0.294·25-s − 0.870·26-s − 0.620·28-s − 1.81·29-s + 0.285·31-s − 0.176·32-s + 1.15·34-s − 1.04·35-s + 0.746·37-s − 0.274·38-s − 0.296·40-s − 0.521·41-s + 1.09·43-s − 0.249·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 1.87T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.43T + 13T^{2} \) |
| 17 | \( 1 + 6.72T + 17T^{2} \) |
| 19 | \( 1 - 1.69T + 19T^{2} \) |
| 23 | \( 1 - 1.69T + 23T^{2} \) |
| 29 | \( 1 + 9.75T + 29T^{2} \) |
| 31 | \( 1 - 1.59T + 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 + 3.33T + 41T^{2} \) |
| 43 | \( 1 - 7.16T + 43T^{2} \) |
| 47 | \( 1 + 4.47T + 47T^{2} \) |
| 53 | \( 1 - 0.184T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 5.75T + 61T^{2} \) |
| 67 | \( 1 + 8.45T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 8.16T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.773653722739241710225384999116, −7.53443665344451862718732616103, −6.83557721123075877429204777068, −6.08131107382062768736264667780, −5.73823406597983189949693834446, −4.31305290953678810042882856556, −3.36746979573927515562215205578, −2.45027557999848578464825754047, −1.45849595295937742726695224659, 0,
1.45849595295937742726695224659, 2.45027557999848578464825754047, 3.36746979573927515562215205578, 4.31305290953678810042882856556, 5.73823406597983189949693834446, 6.08131107382062768736264667780, 6.83557721123075877429204777068, 7.53443665344451862718732616103, 8.773653722739241710225384999116
