| L(s) = 1 | + (2.23 + 0.114i)5-s − 3.07·7-s + 1.12·11-s + 3.92i·13-s + 0.286i·17-s + 5.12i·19-s + (−4.17 + 2.35i)23-s + (4.97 + 0.510i)25-s − 3.70i·29-s − 5.14·31-s + (−6.86 − 0.351i)35-s − 1.46·37-s − 4.70i·41-s − 4.70·43-s − 3.27·47-s + ⋯ |
| L(s) = 1 | + (0.998 + 0.0511i)5-s − 1.16·7-s + 0.338·11-s + 1.08i·13-s + 0.0693i·17-s + 1.17i·19-s + (−0.870 + 0.491i)23-s + (0.994 + 0.102i)25-s − 0.688i·29-s − 0.923·31-s + (−1.16 − 0.0593i)35-s − 0.240·37-s − 0.734i·41-s − 0.716·43-s − 0.477·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7595294601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7595294601\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 - 0.114i)T \) |
| 23 | \( 1 + (4.17 - 2.35i)T \) |
| good | 7 | \( 1 + 3.07T + 7T^{2} \) |
| 11 | \( 1 - 1.12T + 11T^{2} \) |
| 13 | \( 1 - 3.92iT - 13T^{2} \) |
| 17 | \( 1 - 0.286iT - 17T^{2} \) |
| 19 | \( 1 - 5.12iT - 19T^{2} \) |
| 29 | \( 1 + 3.70iT - 29T^{2} \) |
| 31 | \( 1 + 5.14T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 + 4.70iT - 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + 3.27T + 47T^{2} \) |
| 53 | \( 1 + 3.68iT - 53T^{2} \) |
| 59 | \( 1 - 1.30iT - 59T^{2} \) |
| 61 | \( 1 + 5.75iT - 61T^{2} \) |
| 67 | \( 1 + 7.13T + 67T^{2} \) |
| 71 | \( 1 - 1.29iT - 71T^{2} \) |
| 73 | \( 1 + 9.83iT - 73T^{2} \) |
| 79 | \( 1 - 0.954iT - 79T^{2} \) |
| 83 | \( 1 - 7.87iT - 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.961348415738216904025783515099, −8.011882522750786306794254383152, −7.06974133856446914292650118703, −6.40168728788764047313197883203, −5.98337549900670729651790298789, −5.14833201334502235895108722881, −3.99996784844870380333343757555, −3.41565368278689758953895398470, −2.24738670263220336473053881034, −1.53084014402387688728231660013,
0.19908593549117431572226078538, 1.51225794653366531376095975289, 2.71366264941432806881504686571, 3.20666809881502164308405278427, 4.34423262098731502248193845456, 5.31273811015089421998228112606, 5.89905867744322626957588604126, 6.63526087122796260951111868241, 7.14764497795449136122061075522, 8.242315275445173857124511481760
