| L(s) = 1 | − 1.61i·3-s − 5-s + (2.13 + 1.56i)7-s + 0.405·9-s + 6.01·11-s − 4.25·13-s + 1.61i·15-s + 5.42i·17-s + 4.53i·19-s + (2.51 − 3.43i)21-s + 5.19i·23-s + 25-s − 5.48i·27-s − 0.376i·29-s + 2.93·31-s + ⋯ |
| L(s) = 1 | − 0.929i·3-s − 0.447·5-s + (0.806 + 0.590i)7-s + 0.135·9-s + 1.81·11-s − 1.17·13-s + 0.415i·15-s + 1.31i·17-s + 1.04i·19-s + (0.549 − 0.750i)21-s + 1.08i·23-s + 0.200·25-s − 1.05i·27-s − 0.0699i·29-s + 0.527·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.794658050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.794658050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2.13 - 1.56i)T \) |
| good | 3 | \( 1 + 1.61iT - 3T^{2} \) |
| 11 | \( 1 - 6.01T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 - 5.42iT - 17T^{2} \) |
| 19 | \( 1 - 4.53iT - 19T^{2} \) |
| 23 | \( 1 - 5.19iT - 23T^{2} \) |
| 29 | \( 1 + 0.376iT - 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 - 0.372iT - 37T^{2} \) |
| 41 | \( 1 + 5.75iT - 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 - 3.86T + 47T^{2} \) |
| 53 | \( 1 - 10.2iT - 53T^{2} \) |
| 59 | \( 1 + 10.5iT - 59T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 - 0.782T + 67T^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 + 8.77iT - 73T^{2} \) |
| 79 | \( 1 + 8.74iT - 79T^{2} \) |
| 83 | \( 1 - 8.42iT - 83T^{2} \) |
| 89 | \( 1 + 1.94iT - 89T^{2} \) |
| 97 | \( 1 + 3.14iT - 97T^{2} \) |
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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
−9.682775915556463499883858993860, −8.895553702135691543479428187613, −7.971229989648399094511135015849, −7.46208307872979309671237225803, −6.51912765436589663963797388628, −5.76738186660796471272648111038, −4.50570508774496525838043009563, −3.69254006103653457163505793563, −2.06667116487886167684138649917, −1.31879835793098123063851290642,
0.963002119996702278466358610348, 2.65642394420447812638374644738, 4.03197537122952128704134447873, 4.46456481259717450029876233599, 5.19587678691876504417557226090, 6.86463179002311024392260949428, 7.12646981612649247259087846353, 8.342103923215558744123255652298, 9.223529895457069016779131895034, 9.752959832396316494957718024459
