| L(s) = 1 | − 1.13·2-s − 1.70·3-s − 0.703·4-s + 5-s + 1.93·6-s − 4.75·7-s + 3.07·8-s − 0.0975·9-s − 1.13·10-s + 3.46·11-s + 1.19·12-s + 1.17·13-s + 5.41·14-s − 1.70·15-s − 2.09·16-s − 6.20·17-s + 0.111·18-s − 0.703·20-s + 8.10·21-s − 3.93·22-s + 5.05·23-s − 5.24·24-s + 25-s − 1.33·26-s + 5.27·27-s + 3.34·28-s − 1.61·29-s + ⋯ |
| L(s) = 1 | − 0.805·2-s − 0.983·3-s − 0.351·4-s + 0.447·5-s + 0.791·6-s − 1.79·7-s + 1.08·8-s − 0.0325·9-s − 0.360·10-s + 1.04·11-s + 0.346·12-s + 0.324·13-s + 1.44·14-s − 0.439·15-s − 0.524·16-s − 1.50·17-s + 0.0261·18-s − 0.157·20-s + 1.76·21-s − 0.839·22-s + 1.05·23-s − 1.07·24-s + 0.200·25-s − 0.261·26-s + 1.01·27-s + 0.632·28-s − 0.299·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.13T + 2T^{2} \) |
| 3 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 + 4.75T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 + 6.20T + 17T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 + 1.61T + 29T^{2} \) |
| 31 | \( 1 - 7.49T + 31T^{2} \) |
| 37 | \( 1 - 5.98T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 8.06T + 47T^{2} \) |
| 53 | \( 1 - 6.68T + 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 - 6.20T + 61T^{2} \) |
| 67 | \( 1 - 5.62T + 67T^{2} \) |
| 71 | \( 1 + 2.72T + 71T^{2} \) |
| 73 | \( 1 - 3.15T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 8.24T + 83T^{2} \) |
| 89 | \( 1 + 8.83T + 89T^{2} \) |
| 97 | \( 1 + 0.707T + 97T^{2} \) |
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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
−9.008413290271915467619781608174, −8.457953187074868610546336597146, −6.94113184727374495769991084692, −6.59223318398197395373562962525, −5.90800583887361252115700949291, −4.83143647020547364163938877286, −3.92468130988421657728969581405, −2.72768537561347978172728469928, −1.08720259256496384442333576610, 0,
1.08720259256496384442333576610, 2.72768537561347978172728469928, 3.92468130988421657728969581405, 4.83143647020547364163938877286, 5.90800583887361252115700949291, 6.59223318398197395373562962525, 6.94113184727374495769991084692, 8.457953187074868610546336597146, 9.008413290271915467619781608174
