| L(s) = 1 | − 2.52·5-s − i·7-s − 0.148·11-s − 4.12·13-s + 4.48·17-s + 0.775i·19-s + (3.94 + 2.73i)23-s + 1.40·25-s − 4.12i·29-s + 4.31·31-s + 2.52i·35-s + 4.62i·37-s − 1.74i·41-s − 1.88i·43-s − 6.00i·47-s + ⋯ |
| L(s) = 1 | − 1.13·5-s − 0.377i·7-s − 0.0449·11-s − 1.14·13-s + 1.08·17-s + 0.178i·19-s + (0.821 + 0.569i)23-s + 0.280·25-s − 0.765i·29-s + 0.775·31-s + 0.427i·35-s + 0.759i·37-s − 0.271i·41-s − 0.288i·43-s − 0.876i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1323298005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1323298005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 23 | \( 1 + (-3.94 - 2.73i)T \) |
| good | 5 | \( 1 + 2.52T + 5T^{2} \) |
| 11 | \( 1 + 0.148T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 - 4.48T + 17T^{2} \) |
| 19 | \( 1 - 0.775iT - 19T^{2} \) |
| 29 | \( 1 + 4.12iT - 29T^{2} \) |
| 31 | \( 1 - 4.31T + 31T^{2} \) |
| 37 | \( 1 - 4.62iT - 37T^{2} \) |
| 41 | \( 1 + 1.74iT - 41T^{2} \) |
| 43 | \( 1 + 1.88iT - 43T^{2} \) |
| 47 | \( 1 + 6.00iT - 47T^{2} \) |
| 53 | \( 1 - 1.22T + 53T^{2} \) |
| 59 | \( 1 + 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 0.327iT - 61T^{2} \) |
| 67 | \( 1 - 8.77iT - 67T^{2} \) |
| 71 | \( 1 + 9.70iT - 71T^{2} \) |
| 73 | \( 1 + 5.80T + 73T^{2} \) |
| 79 | \( 1 - 11.3iT - 79T^{2} \) |
| 83 | \( 1 - 0.932T + 83T^{2} \) |
| 89 | \( 1 + 7.18T + 89T^{2} \) |
| 97 | \( 1 - 5.77iT - 97T^{2} \) |
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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
−7.78002579617675594524733473553, −7.21190800567846654249069259843, −6.51039705033264537956653951494, −5.42942008422315589474861724925, −4.84270741594144778508493917019, −3.99152949440441240697736996342, −3.37092739398660417308007350825, −2.47484662139893059999746799489, −1.14703210673902340014954431503, −0.04054050426395818109222254612,
1.18125400376745847951742395270, 2.57485498798859718608653944538, 3.15618324268485823917679253766, 4.12827532765090976894616908952, 4.79959440973741920916418348220, 5.48942665500019822970791967761, 6.40214027254679148358690679458, 7.33416866835129821139047873645, 7.59115333141003474554031896978, 8.408933942783188458435531562381
