| L(s) = 1 | − 2.90·3-s + 0.776·5-s − 2.27·7-s + 5.45·9-s + 11-s − 2.17·13-s − 2.25·15-s + 5.46·17-s + 8.46·19-s + 6.61·21-s + 8.47·23-s − 4.39·25-s − 7.15·27-s − 7.09·29-s + 1.35·31-s − 2.90·33-s − 1.76·35-s − 1.08·37-s + 6.32·39-s + 3.64·41-s + 4.94·43-s + 4.23·45-s + 0.962·47-s − 1.82·49-s − 15.8·51-s + 1.67·53-s + 0.776·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s + 0.347·5-s − 0.860·7-s + 1.81·9-s + 0.301·11-s − 0.603·13-s − 0.583·15-s + 1.32·17-s + 1.94·19-s + 1.44·21-s + 1.76·23-s − 0.879·25-s − 1.37·27-s − 1.31·29-s + 0.242·31-s − 0.506·33-s − 0.298·35-s − 0.179·37-s + 1.01·39-s + 0.569·41-s + 0.753·43-s + 0.632·45-s + 0.140·47-s − 0.260·49-s − 2.22·51-s + 0.230·53-s + 0.104·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.041041683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041041683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 - 0.776T + 5T^{2} \) |
| 7 | \( 1 + 2.27T + 7T^{2} \) |
| 13 | \( 1 + 2.17T + 13T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 19 | \( 1 - 8.46T + 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 + 7.09T + 29T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 + 1.08T + 37T^{2} \) |
| 41 | \( 1 - 3.64T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 - 0.962T + 47T^{2} \) |
| 53 | \( 1 - 1.67T + 53T^{2} \) |
| 59 | \( 1 + 3.04T + 59T^{2} \) |
| 61 | \( 1 + 3.49T + 61T^{2} \) |
| 67 | \( 1 - 6.77T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 8.86T + 79T^{2} \) |
| 83 | \( 1 - 2.91T + 83T^{2} \) |
| 89 | \( 1 + 0.376T + 89T^{2} \) |
| 97 | \( 1 - 2.23T + 97T^{2} \) |
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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
−7.58913370182065776302284674999, −7.33702300333662875267353642389, −6.53581868469800846201731617939, −5.75096983896966332158196627503, −5.46232437110615375358189402153, −4.75572372376818584287191572284, −3.66418122605982769592273599189, −2.91420173989206257976952032231, −1.43880668065984725273832308317, −0.63814491340394775656727219325,
0.63814491340394775656727219325, 1.43880668065984725273832308317, 2.91420173989206257976952032231, 3.66418122605982769592273599189, 4.75572372376818584287191572284, 5.46232437110615375358189402153, 5.75096983896966332158196627503, 6.53581868469800846201731617939, 7.33702300333662875267353642389, 7.58913370182065776302284674999
