| L(s) = 1 | − 2.67i·3-s + 4.67i·7-s − 4.14·9-s − 0.672·11-s − 1.14i·13-s − 3.52i·17-s − 5.52·19-s + 12.4·21-s − 3.81i·23-s + 3.05i·27-s + 29-s − 1.52·31-s + 1.79i·33-s − 7.16i·37-s − 3.05·39-s + ⋯ |
| L(s) = 1 | − 1.54i·3-s + 1.76i·7-s − 1.38·9-s − 0.202·11-s − 0.317i·13-s − 0.855i·17-s − 1.26·19-s + 2.72·21-s − 0.795i·23-s + 0.588i·27-s + 0.185·29-s − 0.274·31-s + 0.313i·33-s − 1.17i·37-s − 0.489·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1098266048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1098266048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.67iT - 3T^{2} \) |
| 7 | \( 1 - 4.67iT - 7T^{2} \) |
| 11 | \( 1 + 0.672T + 11T^{2} \) |
| 13 | \( 1 + 1.14iT - 13T^{2} \) |
| 17 | \( 1 + 3.52iT - 17T^{2} \) |
| 19 | \( 1 + 5.52T + 19T^{2} \) |
| 23 | \( 1 + 3.81iT - 23T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + 7.16iT - 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 - 8.96iT - 43T^{2} \) |
| 47 | \( 1 - 6.67iT - 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 7.81iT - 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 + 4.96iT - 73T^{2} \) |
| 79 | \( 1 + 2.38T + 79T^{2} \) |
| 83 | \( 1 - 14.0iT - 83T^{2} \) |
| 89 | \( 1 - 1.63T + 89T^{2} \) |
| 97 | \( 1 - 9.32iT - 97T^{2} \) |
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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
−8.953113208968614624873857441576, −8.075713882577139782989426050700, −7.65981540849304234871004620718, −6.59019743502478560804067157147, −6.13309846109282705300952856754, −5.45787165003908326172696923250, −4.45647266755760237515823561469, −2.73411741217272101909417238836, −2.53817237748939651516678700156, −1.43645747214057803097770483776,
0.03333993842817919259779111983, 1.68083076106882469784198718801, 3.25252600819871416931193780252, 3.90793296450129930832815466672, 4.40334797229183089283781543452, 5.13933422717018452734159081367, 6.21412767776938773074534029443, 6.99181270364461675739222540088, 7.88701510874141454017817595045, 8.616618635124189406701511058708
