| L(s) = 1 | − 0.554·2-s − 3-s − 1.69·4-s + 0.554·6-s + 7-s + 2.04·8-s + 9-s − 0.445·11-s + 1.69·12-s − 0.554·14-s + 2.24·16-s + 3.60·17-s − 0.554·18-s − 4.09·19-s − 21-s + 0.246·22-s − 7.00·23-s − 2.04·24-s − 5·25-s − 27-s − 1.69·28-s + 1.74·29-s + 6.49·31-s − 5.34·32-s + 0.445·33-s − 2·34-s − 1.69·36-s + ⋯ |
| L(s) = 1 | − 0.392·2-s − 0.577·3-s − 0.846·4-s + 0.226·6-s + 0.377·7-s + 0.724·8-s + 0.333·9-s − 0.134·11-s + 0.488·12-s − 0.148·14-s + 0.561·16-s + 0.874·17-s − 0.130·18-s − 0.940·19-s − 0.218·21-s + 0.0526·22-s − 1.46·23-s − 0.418·24-s − 25-s − 0.192·27-s − 0.319·28-s + 0.323·29-s + 1.16·31-s − 0.944·32-s + 0.0774·33-s − 0.342·34-s − 0.282·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.554T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 0.445T + 11T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 + 4.09T + 19T^{2} \) |
| 23 | \( 1 + 7.00T + 23T^{2} \) |
| 29 | \( 1 - 1.74T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 + 7.18T + 37T^{2} \) |
| 41 | \( 1 + 6.89T + 41T^{2} \) |
| 43 | \( 1 - 8.85T + 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 - 9.28T + 53T^{2} \) |
| 59 | \( 1 - 4.98T + 59T^{2} \) |
| 61 | \( 1 + 0.933T + 61T^{2} \) |
| 67 | \( 1 - 1.70T + 67T^{2} \) |
| 71 | \( 1 + 3.80T + 71T^{2} \) |
| 73 | \( 1 - 7.30T + 73T^{2} \) |
| 79 | \( 1 + 0.621T + 79T^{2} \) |
| 83 | \( 1 + 4.27T + 83T^{2} \) |
| 89 | \( 1 - 8.81T + 89T^{2} \) |
| 97 | \( 1 + 6.59T + 97T^{2} \) |
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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
−8.172775883133611591329861693490, −7.70165970117943151204416329535, −6.72202446556602430024158907638, −5.77854997653464892852428672654, −5.26693072721926270555093662862, −4.27527623387980195449518462067, −3.81625613865847604911756365546, −2.30184813324046765327205025885, −1.18233350143246503856732644428, 0,
1.18233350143246503856732644428, 2.30184813324046765327205025885, 3.81625613865847604911756365546, 4.27527623387980195449518462067, 5.26693072721926270555093662862, 5.77854997653464892852428672654, 6.72202446556602430024158907638, 7.70165970117943151204416329535, 8.172775883133611591329861693490
