| L(s) = 1 | − 2-s + 3.25·3-s + 4-s + 2.13·5-s − 3.25·6-s − 4.36·7-s − 8-s + 7.59·9-s − 2.13·10-s − 2.82·11-s + 3.25·12-s + 7.14·13-s + 4.36·14-s + 6.94·15-s + 16-s + 0.270·17-s − 7.59·18-s + 3.73·19-s + 2.13·20-s − 14.2·21-s + 2.82·22-s + 23-s − 3.25·24-s − 0.450·25-s − 7.14·26-s + 14.9·27-s − 4.36·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.87·3-s + 0.5·4-s + 0.953·5-s − 1.32·6-s − 1.64·7-s − 0.353·8-s + 2.53·9-s − 0.674·10-s − 0.851·11-s + 0.939·12-s + 1.98·13-s + 1.16·14-s + 1.79·15-s + 0.250·16-s + 0.0655·17-s − 1.78·18-s + 0.856·19-s + 0.476·20-s − 3.09·21-s + 0.601·22-s + 0.208·23-s − 0.664·24-s − 0.0901·25-s − 1.40·26-s + 2.87·27-s − 0.824·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.350034709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.350034709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 5 | \( 1 - 2.13T + 5T^{2} \) |
| 7 | \( 1 + 4.36T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 7.14T + 13T^{2} \) |
| 17 | \( 1 - 0.270T + 17T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 31 | \( 1 - 1.04T + 31T^{2} \) |
| 37 | \( 1 + 9.76T + 37T^{2} \) |
| 41 | \( 1 - 7.29T + 41T^{2} \) |
| 43 | \( 1 - 8.40T + 43T^{2} \) |
| 47 | \( 1 - 5.26T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 2.01T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 1.86T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 9.50T + 83T^{2} \) |
| 89 | \( 1 - 5.96T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.133915611579944675058982927420, −7.65554243745075161498170661057, −6.76246464813085197517469966840, −6.23602860881148658311924116389, −5.42059570107575812848352883142, −3.89411007023444919997496546532, −3.37520523570689271050620948502, −2.76156708031183640259306477244, −1.99757382167305764732352120208, −1.01275453131712634552545789528,
1.01275453131712634552545789528, 1.99757382167305764732352120208, 2.76156708031183640259306477244, 3.37520523570689271050620948502, 3.89411007023444919997496546532, 5.42059570107575812848352883142, 6.23602860881148658311924116389, 6.76246464813085197517469966840, 7.65554243745075161498170661057, 8.133915611579944675058982927420
