| L(s) = 1 | + 2.23·2-s + 3.00·4-s + 1.23·7-s + 2.23·8-s − 4·11-s − 4.47·13-s + 2.76·14-s − 0.999·16-s − 7.23·17-s + 2.76·19-s − 8.94·22-s + 23-s − 10.0·26-s + 3.70·28-s + 4.47·29-s + 2.47·31-s − 6.70·32-s − 16.1·34-s + 4.47·37-s + 6.18·38-s − 6.94·41-s − 7.70·43-s − 12.0·44-s + 2.23·46-s − 4·47-s − 5.47·49-s − 13.4·52-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.50·4-s + 0.467·7-s + 0.790·8-s − 1.20·11-s − 1.24·13-s + 0.738·14-s − 0.249·16-s − 1.75·17-s + 0.634·19-s − 1.90·22-s + 0.208·23-s − 1.96·26-s + 0.700·28-s + 0.830·29-s + 0.444·31-s − 1.18·32-s − 2.77·34-s + 0.735·37-s + 1.00·38-s − 1.08·41-s − 1.17·43-s − 1.80·44-s + 0.329·46-s − 0.583·47-s − 0.781·49-s − 1.86·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 0.763T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 5.23T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−7.68478877897372664592891740590, −6.86187893238974572079922039142, −6.34348417438734229098535940613, −5.33712232661972509256012598326, −4.77619728132496108747145644917, −4.53891000696561703427098557246, −3.26811208157143016797753365659, −2.65209461058203472766778692329, −1.90726276821360446601291293866, 0,
1.90726276821360446601291293866, 2.65209461058203472766778692329, 3.26811208157143016797753365659, 4.53891000696561703427098557246, 4.77619728132496108747145644917, 5.33712232661972509256012598326, 6.34348417438734229098535940613, 6.86187893238974572079922039142, 7.68478877897372664592891740590
