| L(s) = 1 | − 2.38·2-s + 3-s + 3.69·4-s − 2.38·6-s + 1.19·7-s − 4.03·8-s + 9-s − 2.68·11-s + 3.69·12-s + 5.12·13-s − 2.84·14-s + 2.24·16-s + 0.967·17-s − 2.38·18-s + 1.87·19-s + 1.19·21-s + 6.41·22-s − 7.60·23-s − 4.03·24-s − 12.2·26-s + 27-s + 4.40·28-s − 9.33·29-s − 7.48·31-s + 2.71·32-s − 2.68·33-s − 2.30·34-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.577·3-s + 1.84·4-s − 0.973·6-s + 0.450·7-s − 1.42·8-s + 0.333·9-s − 0.810·11-s + 1.06·12-s + 1.42·13-s − 0.760·14-s + 0.561·16-s + 0.234·17-s − 0.562·18-s + 0.430·19-s + 0.260·21-s + 1.36·22-s − 1.58·23-s − 0.824·24-s − 2.39·26-s + 0.192·27-s + 0.831·28-s − 1.73·29-s − 1.34·31-s + 0.479·32-s − 0.467·33-s − 0.395·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 0.967T + 17T^{2} \) |
| 19 | \( 1 - 1.87T + 19T^{2} \) |
| 23 | \( 1 + 7.60T + 23T^{2} \) |
| 29 | \( 1 + 9.33T + 29T^{2} \) |
| 31 | \( 1 + 7.48T + 31T^{2} \) |
| 37 | \( 1 + 8.35T + 37T^{2} \) |
| 41 | \( 1 - 0.858T + 41T^{2} \) |
| 43 | \( 1 - 6.69T + 43T^{2} \) |
| 53 | \( 1 - 8.42T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 2.45T + 67T^{2} \) |
| 71 | \( 1 + 7.42T + 71T^{2} \) |
| 73 | \( 1 + 5.18T + 73T^{2} \) |
| 79 | \( 1 - 9.80T + 79T^{2} \) |
| 83 | \( 1 - 4.00T + 83T^{2} \) |
| 89 | \( 1 - 0.872T + 89T^{2} \) |
| 97 | \( 1 + 6.63T + 97T^{2} \) |
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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
−8.143897952170585524157523319333, −7.79697424446011538655387571149, −7.21552146661365639547390642221, −6.15118071889874514508163546781, −5.43737350812690985328972502158, −4.08589208469660899357177217079, −3.22289390934708875424656497251, −2.02954870345560608571900726871, −1.47369291531675974339030747766, 0,
1.47369291531675974339030747766, 2.02954870345560608571900726871, 3.22289390934708875424656497251, 4.08589208469660899357177217079, 5.43737350812690985328972502158, 6.15118071889874514508163546781, 7.21552146661365639547390642221, 7.79697424446011538655387571149, 8.143897952170585524157523319333
