| L(s) = 1 | + 1.16·2-s + 3-s − 0.649·4-s − 5-s + 1.16·6-s − 4.28·7-s − 3.07·8-s + 9-s − 1.16·10-s − 0.649·12-s + 5.16·13-s − 4.98·14-s − 15-s − 2.27·16-s + 5·17-s + 1.16·18-s + 5.59·19-s + 0.649·20-s − 4.28·21-s − 0.219·23-s − 3.07·24-s + 25-s + 6.00·26-s + 27-s + 2.78·28-s + 6.41·29-s − 1.16·30-s + ⋯ |
| L(s) = 1 | + 0.821·2-s + 0.577·3-s − 0.324·4-s − 0.447·5-s + 0.474·6-s − 1.62·7-s − 1.08·8-s + 0.333·9-s − 0.367·10-s − 0.187·12-s + 1.43·13-s − 1.33·14-s − 0.258·15-s − 0.569·16-s + 1.21·17-s + 0.273·18-s + 1.28·19-s + 0.145·20-s − 0.935·21-s − 0.0458·23-s − 0.628·24-s + 0.200·25-s + 1.17·26-s + 0.192·27-s + 0.526·28-s + 1.19·29-s − 0.212·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.205857793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.205857793\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 7 | \( 1 + 4.28T + 7T^{2} \) |
| 13 | \( 1 - 5.16T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 5.59T + 19T^{2} \) |
| 23 | \( 1 + 0.219T + 23T^{2} \) |
| 29 | \( 1 - 6.41T + 29T^{2} \) |
| 31 | \( 1 + 2.83T + 31T^{2} \) |
| 37 | \( 1 - 3.92T + 37T^{2} \) |
| 41 | \( 1 - 5.86T + 41T^{2} \) |
| 43 | \( 1 + 8.90T + 43T^{2} \) |
| 47 | \( 1 + 0.237T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + 7.87T + 59T^{2} \) |
| 61 | \( 1 + 4.85T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 9.98T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 8.00T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 2.56T + 89T^{2} \) |
| 97 | \( 1 - 2.01T + 97T^{2} \) |
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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
−9.389496829898253123009357220178, −8.507655335545484977527392241813, −7.74765416797264549187286572595, −6.65934506661259850284450828676, −6.04017727806943301335178090186, −5.17070405973182787312799121381, −3.97850864137795498839847394982, −3.40162607463411160333083533448, −2.94581148335683205809470155411, −0.890671792296941692537083279251,
0.890671792296941692537083279251, 2.94581148335683205809470155411, 3.40162607463411160333083533448, 3.97850864137795498839847394982, 5.17070405973182787312799121381, 6.04017727806943301335178090186, 6.65934506661259850284450828676, 7.74765416797264549187286572595, 8.507655335545484977527392241813, 9.389496829898253123009357220178
