| L(s) = 1 | − 2·2-s + 9.38·3-s + 4·4-s − 18.7·6-s − 8·8-s + 61·9-s + 20·11-s + 37.5·12-s − 65.6·13-s + 16·16-s − 56.2·17-s − 122·18-s + 9.38·19-s − 40·22-s − 48·23-s − 75.0·24-s + 131.·26-s + 318.·27-s − 166·29-s − 206.·31-s − 32·32-s + 187.·33-s + 112.·34-s + 244·36-s + 78·37-s − 18.7·38-s − 616·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.80·3-s + 0.5·4-s − 1.27·6-s − 0.353·8-s + 2.25·9-s + 0.548·11-s + 0.902·12-s − 1.40·13-s + 0.250·16-s − 0.803·17-s − 1.59·18-s + 0.113·19-s − 0.387·22-s − 0.435·23-s − 0.638·24-s + 0.990·26-s + 2.27·27-s − 1.06·29-s − 1.19·31-s − 0.176·32-s + 0.989·33-s + 0.567·34-s + 1.12·36-s + 0.346·37-s − 0.0800·38-s − 2.52·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 9.38T + 27T^{2} \) |
| 11 | \( 1 - 20T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 56.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 9.38T + 6.85e3T^{2} \) |
| 23 | \( 1 + 48T + 1.21e4T^{2} \) |
| 29 | \( 1 + 166T + 2.43e4T^{2} \) |
| 31 | \( 1 + 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 78T + 5.06e4T^{2} \) |
| 41 | \( 1 - 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 436T + 7.95e4T^{2} \) |
| 47 | \( 1 + 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 62T + 1.48e5T^{2} \) |
| 59 | \( 1 + 666.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 272.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 580T + 3.00e5T^{2} \) |
| 71 | \( 1 + 544T + 3.57e5T^{2} \) |
| 73 | \( 1 - 600.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 680T + 4.93e5T^{2} \) |
| 83 | \( 1 + 196.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 656.T + 9.12e5T^{2} \) |
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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
−8.356521496347212196363996474958, −7.38210926602821896058941986305, −7.32298632832326654596333515262, −6.15248300663623477870565263125, −4.79618027490603302055687902302, −3.92586778814333215034839788131, −3.04961695700585160522120906798, −2.22006355507909600984006538865, −1.58269001665792977889560344222, 0,
1.58269001665792977889560344222, 2.22006355507909600984006538865, 3.04961695700585160522120906798, 3.92586778814333215034839788131, 4.79618027490603302055687902302, 6.15248300663623477870565263125, 7.32298632832326654596333515262, 7.38210926602821896058941986305, 8.356521496347212196363996474958
