| L(s) = 1 | + 1.31·2-s − 0.274·4-s − 4.19·7-s − 2.98·8-s + 0.167·11-s + 3.39·13-s − 5.50·14-s − 3.37·16-s + 4.57·17-s + 3.78·19-s + 0.220·22-s + 8.31·23-s + 4.45·26-s + 1.15·28-s − 2.74·29-s − 7.71·31-s + 1.54·32-s + 6.00·34-s − 2.07·37-s + 4.96·38-s + 1.28·41-s − 11.6·43-s − 0.0460·44-s + 10.9·46-s − 9.99·47-s + 10.5·49-s − 0.932·52-s + ⋯ |
| L(s) = 1 | + 0.928·2-s − 0.137·4-s − 1.58·7-s − 1.05·8-s + 0.0505·11-s + 0.941·13-s − 1.47·14-s − 0.843·16-s + 1.10·17-s + 0.867·19-s + 0.0469·22-s + 1.73·23-s + 0.874·26-s + 0.217·28-s − 0.509·29-s − 1.38·31-s + 0.272·32-s + 1.03·34-s − 0.341·37-s + 0.806·38-s + 0.199·41-s − 1.77·43-s − 0.00694·44-s + 1.61·46-s − 1.45·47-s + 1.50·49-s − 0.129·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.31T + 2T^{2} \) |
| 7 | \( 1 + 4.19T + 7T^{2} \) |
| 11 | \( 1 - 0.167T + 11T^{2} \) |
| 13 | \( 1 - 3.39T + 13T^{2} \) |
| 17 | \( 1 - 4.57T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 - 8.31T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 7.71T + 31T^{2} \) |
| 37 | \( 1 + 2.07T + 37T^{2} \) |
| 41 | \( 1 - 1.28T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 9.99T + 47T^{2} \) |
| 53 | \( 1 + 1.07T + 53T^{2} \) |
| 59 | \( 1 - 4.95T + 59T^{2} \) |
| 61 | \( 1 - 2.36T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 8.67T + 73T^{2} \) |
| 79 | \( 1 + 2.64T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 4.06T + 89T^{2} \) |
| 97 | \( 1 - 2.78T + 97T^{2} \) |
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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
−7.59307786841502950384817390127, −6.83627774686781621848007972635, −6.23478173683536203006274778396, −5.51705593350052529528304047591, −5.02465262422360522969916788714, −3.83512322855978992948619913949, −3.34050986367958883944484588318, −2.95728519280046370836826036429, −1.30561839119548029125384698753, 0,
1.30561839119548029125384698753, 2.95728519280046370836826036429, 3.34050986367958883944484588318, 3.83512322855978992948619913949, 5.02465262422360522969916788714, 5.51705593350052529528304047591, 6.23478173683536203006274778396, 6.83627774686781621848007972635, 7.59307786841502950384817390127
