| L(s) = 1 | + 0.342·2-s − 3-s − 1.88·4-s + 5-s − 0.342·6-s − 3.33·7-s − 1.33·8-s + 9-s + 0.342·10-s − 5.45·11-s + 1.88·12-s − 2.23·13-s − 1.14·14-s − 15-s + 3.30·16-s − 1.47·17-s + 0.342·18-s − 1.14·19-s − 1.88·20-s + 3.33·21-s − 1.86·22-s − 6.33·23-s + 1.33·24-s + 25-s − 0.767·26-s − 27-s + 6.28·28-s + ⋯ |
| L(s) = 1 | + 0.242·2-s − 0.577·3-s − 0.941·4-s + 0.447·5-s − 0.139·6-s − 1.26·7-s − 0.470·8-s + 0.333·9-s + 0.108·10-s − 1.64·11-s + 0.543·12-s − 0.621·13-s − 0.305·14-s − 0.258·15-s + 0.827·16-s − 0.358·17-s + 0.0807·18-s − 0.262·19-s − 0.420·20-s + 0.728·21-s − 0.398·22-s − 1.32·23-s + 0.271·24-s + 0.200·25-s − 0.150·26-s − 0.192·27-s + 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02874939334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02874939334\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 0.342T + 2T^{2} \) |
| 7 | \( 1 + 3.33T + 7T^{2} \) |
| 11 | \( 1 + 5.45T + 11T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 + 6.33T + 23T^{2} \) |
| 29 | \( 1 + 0.347T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 - 9.68T + 41T^{2} \) |
| 43 | \( 1 + 6.88T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 6.29T + 53T^{2} \) |
| 59 | \( 1 + 0.834T + 59T^{2} \) |
| 61 | \( 1 - 1.55T + 61T^{2} \) |
| 67 | \( 1 + 2.86T + 67T^{2} \) |
| 71 | \( 1 + 9.96T + 71T^{2} \) |
| 73 | \( 1 + 1.08T + 73T^{2} \) |
| 79 | \( 1 - 6.47T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 5.39T + 89T^{2} \) |
| 97 | \( 1 - 8.77T + 97T^{2} \) |
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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
−8.039919997198112977688445894728, −7.34545281290717880506852099421, −6.44816155959251377665065920682, −5.83225351955474326922554338558, −5.24026795587453098341692902866, −4.63277596494474250589280685509, −3.65844058778088332760502274476, −2.93206470989486049443852628794, −1.90580464229684242964171102881, −0.083113223726094105728854219303,
0.083113223726094105728854219303, 1.90580464229684242964171102881, 2.93206470989486049443852628794, 3.65844058778088332760502274476, 4.63277596494474250589280685509, 5.24026795587453098341692902866, 5.83225351955474326922554338558, 6.44816155959251377665065920682, 7.34545281290717880506852099421, 8.039919997198112977688445894728
