| L(s) = 1 | − 1.71·2-s + 3-s + 0.939·4-s − 1.71·6-s − 0.654·7-s + 1.81·8-s + 9-s + 0.163·11-s + 0.939·12-s + 2.65·13-s + 1.12·14-s − 4.99·16-s − 3.86·17-s − 1.71·18-s − 3.47·19-s − 0.654·21-s − 0.280·22-s + 7.69·23-s + 1.81·24-s − 4.55·26-s + 27-s − 0.614·28-s + 29-s + 5.05·31-s + 4.93·32-s + 0.163·33-s + 6.62·34-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 0.577·3-s + 0.469·4-s − 0.699·6-s − 0.247·7-s + 0.642·8-s + 0.333·9-s + 0.0493·11-s + 0.271·12-s + 0.736·13-s + 0.299·14-s − 1.24·16-s − 0.936·17-s − 0.404·18-s − 0.798·19-s − 0.142·21-s − 0.0597·22-s + 1.60·23-s + 0.371·24-s − 0.892·26-s + 0.192·27-s − 0.116·28-s + 0.185·29-s + 0.907·31-s + 0.871·32-s + 0.0284·33-s + 1.13·34-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
| \(L(1)\) |
\(\approx\) |
\(1.059622693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059622693\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.71T + 2T^{2} \) |
| 7 | \( 1 + 0.654T + 7T^{2} \) |
| 11 | \( 1 - 0.163T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 + 3.86T + 17T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 31 | \( 1 - 5.05T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 6.17T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 4.04T + 53T^{2} \) |
| 59 | \( 1 + 0.328T + 59T^{2} \) |
| 61 | \( 1 + 5.72T + 61T^{2} \) |
| 67 | \( 1 - 3.51T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 1.12T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 7.89T + 83T^{2} \) |
| 89 | \( 1 - 5.04T + 89T^{2} \) |
| 97 | \( 1 - 8.49T + 97T^{2} \) |
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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
−8.909589776036679241387600444557, −8.541462617920048750418482628434, −7.81038706592580911617052081088, −6.88548110999674217911613356232, −6.35152443070820055876835677443, −4.91609492452541702899954662808, −4.16033601284301933421783501531, −3.00962044273387538845704338387, −1.95707845261105360481999170081, −0.804954869493549802355135098687,
0.804954869493549802355135098687, 1.95707845261105360481999170081, 3.00962044273387538845704338387, 4.16033601284301933421783501531, 4.91609492452541702899954662808, 6.35152443070820055876835677443, 6.88548110999674217911613356232, 7.81038706592580911617052081088, 8.541462617920048750418482628434, 8.909589776036679241387600444557
