| L(s) = 1 | + 2.39·2-s + 3-s + 3.72·4-s − 1.40·5-s + 2.39·6-s + 1.96·7-s + 4.13·8-s + 9-s − 3.35·10-s + 6.57·11-s + 3.72·12-s − 13-s + 4.71·14-s − 1.40·15-s + 2.44·16-s − 1.72·17-s + 2.39·18-s + 2.07·19-s − 5.22·20-s + 1.96·21-s + 15.7·22-s + 6.57·23-s + 4.13·24-s − 3.03·25-s − 2.39·26-s + 27-s + 7.34·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 0.577·3-s + 1.86·4-s − 0.627·5-s + 0.977·6-s + 0.744·7-s + 1.46·8-s + 0.333·9-s − 1.06·10-s + 1.98·11-s + 1.07·12-s − 0.277·13-s + 1.26·14-s − 0.361·15-s + 0.610·16-s − 0.417·17-s + 0.564·18-s + 0.475·19-s − 1.16·20-s + 0.429·21-s + 3.35·22-s + 1.37·23-s + 0.844·24-s − 0.606·25-s − 0.469·26-s + 0.192·27-s + 1.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.197820458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.197820458\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 5 | \( 1 + 1.40T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 - 6.57T + 11T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 19 | \( 1 - 2.07T + 19T^{2} \) |
| 23 | \( 1 - 6.57T + 23T^{2} \) |
| 29 | \( 1 - 0.697T + 29T^{2} \) |
| 31 | \( 1 - 1.51T + 31T^{2} \) |
| 37 | \( 1 + 8.95T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 7.65T + 43T^{2} \) |
| 47 | \( 1 + 5.50T + 47T^{2} \) |
| 53 | \( 1 - 6.02T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 - 4.09T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 7.11T + 79T^{2} \) |
| 83 | \( 1 - 4.05T + 83T^{2} \) |
| 89 | \( 1 - 3.74T + 89T^{2} \) |
| 97 | \( 1 - 2.19T + 97T^{2} \) |
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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
−8.395104070430086836250090315390, −7.38708889395794880140258136628, −6.86907590682962866592993655876, −6.23599676341953818319290253535, −5.07104574431776160926811932761, −4.67459494574591991462470242541, −3.69247210559910595690300765867, −3.50015667218892676838638775624, −2.26191791295069899051582157409, −1.34170984195428247365406578682,
1.34170984195428247365406578682, 2.26191791295069899051582157409, 3.50015667218892676838638775624, 3.69247210559910595690300765867, 4.67459494574591991462470242541, 5.07104574431776160926811932761, 6.23599676341953818319290253535, 6.86907590682962866592993655876, 7.38708889395794880140258136628, 8.395104070430086836250090315390
