| L(s) = 1 | − 0.317i·5-s − 1.44i·7-s + 1.09·11-s − 2.89·13-s − 3.46i·17-s + 4.89i·19-s + 2.82·23-s + 4.89·25-s − 9.12i·29-s − 7.44i·31-s − 0.460·35-s − 4.89·37-s − 9.75i·41-s + 6.89i·43-s − 9.12·47-s + ⋯ |
| L(s) = 1 | − 0.142i·5-s − 0.547i·7-s + 0.330·11-s − 0.804·13-s − 0.840i·17-s + 1.12i·19-s + 0.589·23-s + 0.979·25-s − 1.69i·29-s − 1.33i·31-s − 0.0778·35-s − 0.805·37-s − 1.52i·41-s + 1.05i·43-s − 1.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.363479697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.363479697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.317iT - 5T^{2} \) |
| 7 | \( 1 + 1.44iT - 7T^{2} \) |
| 11 | \( 1 - 1.09T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 9.12iT - 29T^{2} \) |
| 31 | \( 1 + 7.44iT - 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 9.75iT - 41T^{2} \) |
| 43 | \( 1 - 6.89iT - 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 + 4.41iT - 53T^{2} \) |
| 59 | \( 1 + 9.12T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 5.10iT - 67T^{2} \) |
| 71 | \( 1 - 7.56T + 71T^{2} \) |
| 73 | \( 1 - 1.89T + 73T^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 11.9iT - 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−9.303559282348069994017498776353, −8.226889396218829029113350494867, −7.55162682111350830522495621778, −6.81665015127744345983934365911, −5.90083940782119082330652527223, −4.93242004473959737927264825101, −4.16835175865733896843310382648, −3.13933296115095767517559616887, −1.96958759958764648889852690180, −0.53299476197290196490512769634,
1.37706470852213439792547562588, 2.65466276443996398665011776040, 3.45398995811420679498994638287, 4.81467098180068386576103182849, 5.23937428807383987291611273361, 6.56431280351604046835907917137, 6.92263869672202396305037888579, 7.998767567836365882664195185164, 8.940254043545858080850180668166, 9.227962935585059355535013677923
