| L(s) = 1 | + 2.55·2-s + 4.52·4-s + 3.88·5-s + 2.95·7-s + 6.45·8-s + 9.91·10-s − 0.599·11-s − 3.25·13-s + 7.54·14-s + 7.44·16-s − 0.0920·17-s − 3.04·19-s + 17.5·20-s − 1.53·22-s + 23-s + 10.0·25-s − 8.32·26-s + 13.3·28-s − 29-s − 7.11·31-s + 6.10·32-s − 0.235·34-s + 11.4·35-s + 1.16·37-s − 7.78·38-s + 25.0·40-s + 1.64·41-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.26·4-s + 1.73·5-s + 1.11·7-s + 2.28·8-s + 3.13·10-s − 0.180·11-s − 0.903·13-s + 2.01·14-s + 1.86·16-s − 0.0223·17-s − 0.698·19-s + 3.92·20-s − 0.326·22-s + 0.208·23-s + 2.01·25-s − 1.63·26-s + 2.52·28-s − 0.185·29-s − 1.27·31-s + 1.07·32-s − 0.0403·34-s + 1.93·35-s + 0.191·37-s − 1.26·38-s + 3.96·40-s + 0.256·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.745568170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.745568170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.55T + 2T^{2} \) |
| 5 | \( 1 - 3.88T + 5T^{2} \) |
| 7 | \( 1 - 2.95T + 7T^{2} \) |
| 11 | \( 1 + 0.599T + 11T^{2} \) |
| 13 | \( 1 + 3.25T + 13T^{2} \) |
| 17 | \( 1 + 0.0920T + 17T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 31 | \( 1 + 7.11T + 31T^{2} \) |
| 37 | \( 1 - 1.16T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 + 0.124T + 43T^{2} \) |
| 47 | \( 1 - 2.86T + 47T^{2} \) |
| 53 | \( 1 - 5.99T + 53T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 - 7.62T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 + 0.673T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 4.26T + 79T^{2} \) |
| 83 | \( 1 + 0.758T + 83T^{2} \) |
| 89 | \( 1 - 4.20T + 89T^{2} \) |
| 97 | \( 1 + 3.47T + 97T^{2} \) |
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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
−7.77747263432990565810138017366, −7.01373716097737534608478336952, −6.42126297954769476626163710737, −5.54693286040629767724224965268, −5.32189812991662064325981500864, −4.65858632766812614699618184762, −3.84573185259369273367057005552, −2.63904854497381127774416602064, −2.21364242164700353140896684371, −1.49206502426777685669249697170,
1.49206502426777685669249697170, 2.21364242164700353140896684371, 2.63904854497381127774416602064, 3.84573185259369273367057005552, 4.65858632766812614699618184762, 5.32189812991662064325981500864, 5.54693286040629767724224965268, 6.42126297954769476626163710737, 7.01373716097737534608478336952, 7.77747263432990565810138017366
