| L(s) = 1 | + 2.44·2-s + 3.98·4-s − 3.00·5-s − 3.89·7-s + 4.84·8-s − 7.34·10-s + 0.115·13-s − 9.51·14-s + 3.88·16-s + 1.12·17-s − 1.57·19-s − 11.9·20-s − 2.49·23-s + 4.01·25-s + 0.283·26-s − 15.4·28-s + 4.11·29-s − 1.63·31-s − 0.193·32-s + 2.74·34-s + 11.6·35-s + 5.39·37-s − 3.85·38-s − 14.5·40-s − 5.06·41-s + 8.08·43-s − 6.10·46-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 1.99·4-s − 1.34·5-s − 1.47·7-s + 1.71·8-s − 2.32·10-s + 0.0321·13-s − 2.54·14-s + 0.970·16-s + 0.272·17-s − 0.362·19-s − 2.67·20-s − 0.520·23-s + 0.803·25-s + 0.0555·26-s − 2.92·28-s + 0.764·29-s − 0.292·31-s − 0.0341·32-s + 0.470·34-s + 1.97·35-s + 0.887·37-s − 0.626·38-s − 2.29·40-s − 0.791·41-s + 1.23·43-s − 0.899·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.172035610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.172035610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 5 | \( 1 + 3.00T + 5T^{2} \) |
| 7 | \( 1 + 3.89T + 7T^{2} \) |
| 13 | \( 1 - 0.115T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 1.57T + 19T^{2} \) |
| 23 | \( 1 + 2.49T + 23T^{2} \) |
| 29 | \( 1 - 4.11T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 - 5.39T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 - 8.08T + 43T^{2} \) |
| 47 | \( 1 - 3.08T + 47T^{2} \) |
| 53 | \( 1 - 8.14T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 2.41T + 61T^{2} \) |
| 67 | \( 1 + 6.34T + 67T^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 8.22T + 79T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 + 4.11T + 89T^{2} \) |
| 97 | \( 1 + 2.58T + 97T^{2} \) |
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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
−7.26097559462684831670371192872, −6.91354814842086717540432005753, −6.12265071663642261589120608391, −5.67615536842115726844119727101, −4.68443170016937752204049367532, −4.07394604287643352210665533961, −3.61544398631093406299572778017, −3.00882033656480243982614304309, −2.26399827008312543451175774534, −0.61315665661949971652943402757,
0.61315665661949971652943402757, 2.26399827008312543451175774534, 3.00882033656480243982614304309, 3.61544398631093406299572778017, 4.07394604287643352210665533961, 4.68443170016937752204049367532, 5.67615536842115726844119727101, 6.12265071663642261589120608391, 6.91354814842086717540432005753, 7.26097559462684831670371192872
