| L(s) = 1 | + 2.41·2-s + 3.82·4-s − 5-s + 4.41·8-s − 2.41·10-s − 0.828·11-s + 4.82·13-s + 2.99·16-s + 3.65·17-s − 2.82·19-s − 3.82·20-s − 1.99·22-s + 7.65·23-s + 25-s + 11.6·26-s + 8·29-s + 8.48·31-s − 1.58·32-s + 8.82·34-s − 6·37-s − 6.82·38-s − 4.41·40-s − 3.65·41-s − 9.65·43-s − 3.17·44-s + 18.4·46-s + 4·47-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s − 0.447·5-s + 1.56·8-s − 0.763·10-s − 0.249·11-s + 1.33·13-s + 0.749·16-s + 0.886·17-s − 0.648·19-s − 0.856·20-s − 0.426·22-s + 1.59·23-s + 0.200·25-s + 2.28·26-s + 1.48·29-s + 1.52·31-s − 0.280·32-s + 1.51·34-s − 0.986·37-s − 1.10·38-s − 0.697·40-s − 0.571·41-s − 1.47·43-s − 0.478·44-s + 2.72·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.970089914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.970089914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 7.31T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−8.754125663964176212103119716902, −8.269439050647145101655782950637, −7.07771421492104366693541276644, −6.57463225014792953314705973824, −5.72337611636478613943790620607, −4.96365759228362046422209983298, −4.22997828987748240978002708109, −3.36194828744898197018277608155, −2.76028544630804318875566575926, −1.26057609698056513359560593959,
1.26057609698056513359560593959, 2.76028544630804318875566575926, 3.36194828744898197018277608155, 4.22997828987748240978002708109, 4.96365759228362046422209983298, 5.72337611636478613943790620607, 6.57463225014792953314705973824, 7.07771421492104366693541276644, 8.269439050647145101655782950637, 8.754125663964176212103119716902
