| L(s) = 1 | − 0.994·2-s − 2.24·3-s − 1.01·4-s + 0.441·5-s + 2.22·6-s − 0.726·7-s + 2.99·8-s + 2.02·9-s − 0.438·10-s + 1.60·11-s + 2.26·12-s + 0.722·14-s − 0.989·15-s − 0.955·16-s + 4.94·17-s − 2.01·18-s + 7.16·19-s − 0.446·20-s + 1.62·21-s − 1.59·22-s + 8.21·23-s − 6.71·24-s − 4.80·25-s + 2.19·27-s + 0.734·28-s + 5.88·29-s + 0.983·30-s + ⋯ |
| L(s) = 1 | − 0.703·2-s − 1.29·3-s − 0.505·4-s + 0.197·5-s + 0.909·6-s − 0.274·7-s + 1.05·8-s + 0.674·9-s − 0.138·10-s + 0.483·11-s + 0.654·12-s + 0.193·14-s − 0.255·15-s − 0.238·16-s + 1.19·17-s − 0.474·18-s + 1.64·19-s − 0.0997·20-s + 0.355·21-s − 0.340·22-s + 1.71·23-s − 1.36·24-s − 0.961·25-s + 0.421·27-s + 0.138·28-s + 1.09·29-s + 0.179·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9041282067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9041282067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.994T + 2T^{2} \) |
| 3 | \( 1 + 2.24T + 3T^{2} \) |
| 5 | \( 1 - 0.441T + 5T^{2} \) |
| 7 | \( 1 + 0.726T + 7T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 17 | \( 1 - 4.94T + 17T^{2} \) |
| 19 | \( 1 - 7.16T + 19T^{2} \) |
| 23 | \( 1 - 8.21T + 23T^{2} \) |
| 29 | \( 1 - 5.88T + 29T^{2} \) |
| 37 | \( 1 - 6.46T + 37T^{2} \) |
| 41 | \( 1 - 2.09T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 2.12T + 47T^{2} \) |
| 53 | \( 1 - 4.51T + 53T^{2} \) |
| 59 | \( 1 + 5.07T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 8.55T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 1.20T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 6.96T + 97T^{2} \) |
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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
−8.181731319361392695666043542704, −7.46860901559868977577377894193, −6.83774688588279426801659401268, −5.92241866676062463464543337528, −5.37674865730428422138539757346, −4.77066902101103191562278599097, −3.83088129995670335490256425458, −2.82330921557508752807100120030, −1.14799928111854251316777891423, −0.799368969696201930904440732818,
0.799368969696201930904440732818, 1.14799928111854251316777891423, 2.82330921557508752807100120030, 3.83088129995670335490256425458, 4.77066902101103191562278599097, 5.37674865730428422138539757346, 5.92241866676062463464543337528, 6.83774688588279426801659401268, 7.46860901559868977577377894193, 8.181731319361392695666043542704
