| L(s) = 1 | − 0.329·2-s + 3-s − 1.89·4-s + 2.08·5-s − 0.329·6-s − 4.32·7-s + 1.28·8-s + 9-s − 0.687·10-s − 11-s − 1.89·12-s + 2.30·13-s + 1.42·14-s + 2.08·15-s + 3.36·16-s − 0.847·17-s − 0.329·18-s − 0.842·19-s − 3.94·20-s − 4.32·21-s + 0.329·22-s + 4.59·23-s + 1.28·24-s − 0.640·25-s − 0.758·26-s + 27-s + 8.17·28-s + ⋯ |
| L(s) = 1 | − 0.232·2-s + 0.577·3-s − 0.945·4-s + 0.933·5-s − 0.134·6-s − 1.63·7-s + 0.453·8-s + 0.333·9-s − 0.217·10-s − 0.301·11-s − 0.546·12-s + 0.639·13-s + 0.380·14-s + 0.539·15-s + 0.840·16-s − 0.205·17-s − 0.0776·18-s − 0.193·19-s − 0.883·20-s − 0.943·21-s + 0.0702·22-s + 0.957·23-s + 0.261·24-s − 0.128·25-s − 0.148·26-s + 0.192·27-s + 1.54·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.455687551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455687551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.329T + 2T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 + 4.32T + 7T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 + 0.847T + 17T^{2} \) |
| 19 | \( 1 + 0.842T + 19T^{2} \) |
| 23 | \( 1 - 4.59T + 23T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 2.01T + 41T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 + 0.441T + 47T^{2} \) |
| 53 | \( 1 + 5.16T + 53T^{2} \) |
| 59 | \( 1 - 7.75T + 59T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 - 2.20T + 73T^{2} \) |
| 79 | \( 1 - 5.20T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 0.539T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−9.355807194865997892745843109976, −8.600014441914657479406062800530, −7.76298366028874157884231795966, −6.69774948252449264930529457893, −6.06024400385255405497395081015, −5.18656201225552419278095617082, −4.06149347841859495606140443052, −3.28871321288934316379164937505, −2.31853447013109704192225570227, −0.815911106883510745942186120656,
0.815911106883510745942186120656, 2.31853447013109704192225570227, 3.28871321288934316379164937505, 4.06149347841859495606140443052, 5.18656201225552419278095617082, 6.06024400385255405497395081015, 6.69774948252449264930529457893, 7.76298366028874157884231795966, 8.600014441914657479406062800530, 9.355807194865997892745843109976
