| L(s) = 1 | − 2.69·2-s + 5.23·4-s + 3.51·5-s + 7-s − 8.70·8-s − 9.44·10-s + 3.78·11-s + 1.24·13-s − 2.69·14-s + 12.9·16-s − 5.98·17-s − 3.38·19-s + 18.3·20-s − 10.1·22-s + 23-s + 7.32·25-s − 3.33·26-s + 5.23·28-s + 7.02·29-s + 6.26·31-s − 17.4·32-s + 16.1·34-s + 3.51·35-s + 4.84·37-s + 9.09·38-s − 30.5·40-s + 1.78·41-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.61·4-s + 1.57·5-s + 0.377·7-s − 3.07·8-s − 2.98·10-s + 1.14·11-s + 0.344·13-s − 0.718·14-s + 3.23·16-s − 1.45·17-s − 0.775·19-s + 4.11·20-s − 2.17·22-s + 0.208·23-s + 1.46·25-s − 0.654·26-s + 0.989·28-s + 1.30·29-s + 1.12·31-s − 3.08·32-s + 2.76·34-s + 0.593·35-s + 0.795·37-s + 1.47·38-s − 4.83·40-s + 0.278·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.089800916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.089800916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 + 5.98T + 17T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 29 | \( 1 - 7.02T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 4.84T + 37T^{2} \) |
| 41 | \( 1 - 1.78T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 + 3.90T + 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 - 2.89T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 + 7.02T + 83T^{2} \) |
| 89 | \( 1 - 1.59T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−9.480952951866280132691606666180, −8.761415041535380770678433884865, −8.399466212566157470174904951290, −7.10083897756209394677788441741, −6.39412804728287584207614565394, −6.05084602583014467681896544741, −4.51366669472755673491289283218, −2.71947553984394305517198844332, −1.94024711811846161341161892521, −1.06067370396726172783507812074,
1.06067370396726172783507812074, 1.94024711811846161341161892521, 2.71947553984394305517198844332, 4.51366669472755673491289283218, 6.05084602583014467681896544741, 6.39412804728287584207614565394, 7.10083897756209394677788441741, 8.399466212566157470174904951290, 8.761415041535380770678433884865, 9.480952951866280132691606666180
