L(s) = 1 | + 2.34·2-s + 3-s + 3.48·4-s − 0.853·5-s + 2.34·6-s + 3.48·8-s + 9-s − 2·10-s − 3.34·11-s + 3.48·12-s − 3.14·13-s − 0.853·15-s + 1.19·16-s − 3.63·17-s + 2.34·18-s − 1.14·19-s − 2.97·20-s − 7.83·22-s − 2.85·23-s + 3.48·24-s − 4.27·25-s − 7.37·26-s + 27-s − 8.02·29-s − 2·30-s − 9.86·31-s − 4.17·32-s + ⋯ |
L(s) = 1 | + 1.65·2-s + 0.577·3-s + 1.74·4-s − 0.381·5-s + 0.956·6-s + 1.23·8-s + 0.333·9-s − 0.632·10-s − 1.00·11-s + 1.00·12-s − 0.872·13-s − 0.220·15-s + 0.299·16-s − 0.881·17-s + 0.552·18-s − 0.262·19-s − 0.666·20-s − 1.66·22-s − 0.595·23-s + 0.712·24-s − 0.854·25-s − 1.44·26-s + 0.192·27-s − 1.49·29-s − 0.365·30-s − 1.77·31-s − 0.738·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 5 | \( 1 + 0.853T + 5T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 + 3.63T + 17T^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 + 9.86T + 31T^{2} \) |
| 37 | \( 1 - 8.19T + 37T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 8.32T + 47T^{2} \) |
| 53 | \( 1 - 1.60T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 4.19T + 61T^{2} \) |
| 67 | \( 1 - 6.10T + 67T^{2} \) |
| 71 | \( 1 + 8.65T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 3.60T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 - 7.53T + 97T^{2} \) |
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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
−7.51584899694020341119861063282, −7.03435185340450888653458762601, −6.03705165640759462958593198826, −5.44925842869473204003440256971, −4.71941309232517034831688769665, −3.99814127938129766150250422065, −3.50317544407652326755331625285, −2.34783463452892786753153583100, −2.16916735838813378089131663799, 0,
2.16916735838813378089131663799, 2.34783463452892786753153583100, 3.50317544407652326755331625285, 3.99814127938129766150250422065, 4.71941309232517034831688769665, 5.44925842869473204003440256971, 6.03705165640759462958593198826, 7.03435185340450888653458762601, 7.51584899694020341119861063282