| L(s) = 1 | + 2-s + 3-s + 4-s − 3.26·5-s + 6-s + 1.36·7-s + 8-s + 9-s − 3.26·10-s + 0.184·11-s + 12-s − 13-s + 1.36·14-s − 3.26·15-s + 16-s − 2.62·17-s + 18-s + 0.790·19-s − 3.26·20-s + 1.36·21-s + 0.184·22-s + 5.78·23-s + 24-s + 5.67·25-s − 26-s + 27-s + 1.36·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.46·5-s + 0.408·6-s + 0.517·7-s + 0.353·8-s + 0.333·9-s − 1.03·10-s + 0.0555·11-s + 0.288·12-s − 0.277·13-s + 0.365·14-s − 0.843·15-s + 0.250·16-s − 0.637·17-s + 0.235·18-s + 0.181·19-s − 0.730·20-s + 0.298·21-s + 0.0392·22-s + 1.20·23-s + 0.204·24-s + 1.13·25-s − 0.196·26-s + 0.192·27-s + 0.258·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.190702669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.190702669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 3.26T + 5T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 - 0.184T + 11T^{2} \) |
| 17 | \( 1 + 2.62T + 17T^{2} \) |
| 19 | \( 1 - 0.790T + 19T^{2} \) |
| 23 | \( 1 - 5.78T + 23T^{2} \) |
| 29 | \( 1 - 4.28T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 7.71T + 37T^{2} \) |
| 41 | \( 1 - 5.89T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 + 0.567T + 47T^{2} \) |
| 53 | \( 1 - 5.87T + 53T^{2} \) |
| 59 | \( 1 - 0.213T + 59T^{2} \) |
| 61 | \( 1 - 0.462T + 61T^{2} \) |
| 67 | \( 1 - 8.44T + 67T^{2} \) |
| 71 | \( 1 - 1.02T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 8.60T + 79T^{2} \) |
| 83 | \( 1 + 7.46T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 9.07T + 97T^{2} \) |
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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
−7.78048107638675623482075285867, −7.18467108148751214597548158453, −6.67372239048303619381574894844, −5.55309450812202130476358113061, −4.78073717114317829208325380509, −4.26582016127053283939421828315, −3.59078446668488018252904725671, −2.90463246217951931302867761273, −2.00197895392867314263226073137, −0.76945833045029498477677712178,
0.76945833045029498477677712178, 2.00197895392867314263226073137, 2.90463246217951931302867761273, 3.59078446668488018252904725671, 4.26582016127053283939421828315, 4.78073717114317829208325380509, 5.55309450812202130476358113061, 6.67372239048303619381574894844, 7.18467108148751214597548158453, 7.78048107638675623482075285867
