| L(s) = 1 | − 2-s − 0.478·3-s + 4-s − 0.306·5-s + 0.478·6-s − 3.45·7-s − 8-s − 2.77·9-s + 0.306·10-s + 4.56·11-s − 0.478·12-s − 2.40·13-s + 3.45·14-s + 0.146·15-s + 16-s + 5.75·17-s + 2.77·18-s − 1.33·19-s − 0.306·20-s + 1.65·21-s − 4.56·22-s − 23-s + 0.478·24-s − 4.90·25-s + 2.40·26-s + 2.76·27-s − 3.45·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.276·3-s + 0.5·4-s − 0.137·5-s + 0.195·6-s − 1.30·7-s − 0.353·8-s − 0.923·9-s + 0.0970·10-s + 1.37·11-s − 0.138·12-s − 0.667·13-s + 0.922·14-s + 0.0379·15-s + 0.250·16-s + 1.39·17-s + 0.653·18-s − 0.307·19-s − 0.0686·20-s + 0.360·21-s − 0.972·22-s − 0.208·23-s + 0.0976·24-s − 0.981·25-s + 0.472·26-s + 0.531·27-s − 0.652·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6637219754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6637219754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 0.478T + 3T^{2} \) |
| 5 | \( 1 + 0.306T + 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 11 | \( 1 - 4.56T + 11T^{2} \) |
| 13 | \( 1 + 2.40T + 13T^{2} \) |
| 17 | \( 1 - 5.75T + 17T^{2} \) |
| 19 | \( 1 + 1.33T + 19T^{2} \) |
| 29 | \( 1 + 0.273T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 + 6.26T + 37T^{2} \) |
| 41 | \( 1 - 0.926T + 41T^{2} \) |
| 43 | \( 1 + 5.74T + 43T^{2} \) |
| 47 | \( 1 + 7.11T + 47T^{2} \) |
| 53 | \( 1 - 1.89T + 53T^{2} \) |
| 59 | \( 1 - 2.32T + 59T^{2} \) |
| 61 | \( 1 - 2.06T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 1.77T + 97T^{2} \) |
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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
−8.143044001502978690294492089046, −7.38345230066220816848116248889, −6.55912556020814546143291483118, −6.19735381873033204074098885665, −5.47959120404138838814809599015, −4.35822044442748528125101213757, −3.35958287528485830885297015802, −2.92118152581468902080117094544, −1.65044800676374561543792770882, −0.48018026652780798473303035721,
0.48018026652780798473303035721, 1.65044800676374561543792770882, 2.92118152581468902080117094544, 3.35958287528485830885297015802, 4.35822044442748528125101213757, 5.47959120404138838814809599015, 6.19735381873033204074098885665, 6.55912556020814546143291483118, 7.38345230066220816848116248889, 8.143044001502978690294492089046
