| L(s) = 1 | − 2.15·2-s − 3-s + 2.63·4-s + 2.15·6-s − 1.51·7-s − 1.37·8-s + 9-s + 1.88·11-s − 2.63·12-s − 0.484·13-s + 3.26·14-s − 2.31·16-s − 3.39·17-s − 2.15·18-s + 2.85·19-s + 1.51·21-s − 4.06·22-s + 1.28·23-s + 1.37·24-s + 1.04·26-s − 27-s − 3.99·28-s + 29-s − 5.47·31-s + 7.73·32-s − 1.88·33-s + 7.30·34-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 0.577·3-s + 1.31·4-s + 0.879·6-s − 0.572·7-s − 0.485·8-s + 0.333·9-s + 0.569·11-s − 0.761·12-s − 0.134·13-s + 0.872·14-s − 0.579·16-s − 0.822·17-s − 0.507·18-s + 0.654·19-s + 0.330·21-s − 0.867·22-s + 0.267·23-s + 0.280·24-s + 0.204·26-s − 0.192·27-s − 0.755·28-s + 0.185·29-s − 0.983·31-s + 1.36·32-s − 0.328·33-s + 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 11 | \( 1 - 1.88T + 11T^{2} \) |
| 13 | \( 1 + 0.484T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 - 2.85T + 19T^{2} \) |
| 23 | \( 1 - 1.28T + 23T^{2} \) |
| 31 | \( 1 + 5.47T + 31T^{2} \) |
| 37 | \( 1 + 7.43T + 37T^{2} \) |
| 41 | \( 1 - 8.28T + 41T^{2} \) |
| 43 | \( 1 + 1.43T + 43T^{2} \) |
| 47 | \( 1 - 2.25T + 47T^{2} \) |
| 53 | \( 1 - 5.62T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 7.40T + 61T^{2} \) |
| 67 | \( 1 - 8.76T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 2.80T + 73T^{2} \) |
| 79 | \( 1 + 9.34T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 - 7.24T + 89T^{2} \) |
| 97 | \( 1 + 4.59T + 97T^{2} \) |
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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
−8.899530275508883361238853153598, −8.008704174936230266392328069000, −7.09870259799149147723199774222, −6.72750088290855121381083325849, −5.76210118054371517262020054989, −4.71367160765961938200506968206, −3.62075680366259142733934451502, −2.30666109723907007123194318256, −1.18436371530917250460372208290, 0,
1.18436371530917250460372208290, 2.30666109723907007123194318256, 3.62075680366259142733934451502, 4.71367160765961938200506968206, 5.76210118054371517262020054989, 6.72750088290855121381083325849, 7.09870259799149147723199774222, 8.008704174936230266392328069000, 8.899530275508883361238853153598
