| L(s) = 1 | + 2-s + 2.55·3-s + 4-s − 1.03·5-s + 2.55·6-s + 2.77·7-s + 8-s + 3.52·9-s − 1.03·10-s + 11-s + 2.55·12-s + 2.63·13-s + 2.77·14-s − 2.63·15-s + 16-s − 5.95·17-s + 3.52·18-s − 1.03·20-s + 7.08·21-s + 22-s + 2.55·23-s + 2.55·24-s − 3.93·25-s + 2.63·26-s + 1.34·27-s + 2.77·28-s + 2.14·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.47·3-s + 0.5·4-s − 0.461·5-s + 1.04·6-s + 1.04·7-s + 0.353·8-s + 1.17·9-s − 0.326·10-s + 0.301·11-s + 0.737·12-s + 0.731·13-s + 0.741·14-s − 0.680·15-s + 0.250·16-s − 1.44·17-s + 0.831·18-s − 0.230·20-s + 1.54·21-s + 0.213·22-s + 0.533·23-s + 0.521·24-s − 0.787·25-s + 0.517·26-s + 0.259·27-s + 0.524·28-s + 0.397·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.442890314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.442890314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.55T + 3T^{2} \) |
| 5 | \( 1 + 1.03T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 13 | \( 1 - 2.63T + 13T^{2} \) |
| 17 | \( 1 + 5.95T + 17T^{2} \) |
| 23 | \( 1 - 2.55T + 23T^{2} \) |
| 29 | \( 1 - 2.14T + 29T^{2} \) |
| 31 | \( 1 + 0.213T + 31T^{2} \) |
| 37 | \( 1 - 7.84T + 37T^{2} \) |
| 41 | \( 1 - 9.78T + 41T^{2} \) |
| 43 | \( 1 - 3.36T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 4.03T + 53T^{2} \) |
| 59 | \( 1 + 3.90T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + 2.41T + 67T^{2} \) |
| 71 | \( 1 - 3.00T + 71T^{2} \) |
| 73 | \( 1 - 0.980T + 73T^{2} \) |
| 79 | \( 1 - 9.39T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 7.37T + 97T^{2} \) |
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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
−7.76435012223711706062602009497, −7.43584410776283043267134472653, −6.47225214335536116114704640512, −5.75647360842118595995463595666, −4.64160784728556287845967552538, −4.21163160197716598272375178712, −3.64422290323085891971291829110, −2.64479701902380552011670169060, −2.14722450120192038704259033741, −1.12897691040664537519156137224,
1.12897691040664537519156137224, 2.14722450120192038704259033741, 2.64479701902380552011670169060, 3.64422290323085891971291829110, 4.21163160197716598272375178712, 4.64160784728556287845967552538, 5.75647360842118595995463595666, 6.47225214335536116114704640512, 7.43584410776283043267134472653, 7.76435012223711706062602009497
