| L(s) = 1 | + 2-s − 3.33·3-s + 4-s − 3.73·5-s − 3.33·6-s + 2.28·7-s + 8-s + 8.09·9-s − 3.73·10-s − 6.21·11-s − 3.33·12-s − 6.33·13-s + 2.28·14-s + 12.4·15-s + 16-s − 7.48·17-s + 8.09·18-s + 3.49·19-s − 3.73·20-s − 7.59·21-s − 6.21·22-s + 23-s − 3.33·24-s + 8.93·25-s − 6.33·26-s − 16.9·27-s + 2.28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.92·3-s + 0.5·4-s − 1.66·5-s − 1.35·6-s + 0.862·7-s + 0.353·8-s + 2.69·9-s − 1.18·10-s − 1.87·11-s − 0.961·12-s − 1.75·13-s + 0.609·14-s + 3.21·15-s + 0.250·16-s − 1.81·17-s + 1.90·18-s + 0.801·19-s − 0.834·20-s − 1.65·21-s − 1.32·22-s + 0.208·23-s − 0.679·24-s + 1.78·25-s − 1.24·26-s − 3.26·27-s + 0.431·28-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
| \(L(1)\) |
\(\approx\) |
\(0.01249126706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01249126706\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.33T + 3T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 + 6.21T + 11T^{2} \) |
| 13 | \( 1 + 6.33T + 13T^{2} \) |
| 17 | \( 1 + 7.48T + 17T^{2} \) |
| 19 | \( 1 - 3.49T + 19T^{2} \) |
| 29 | \( 1 + 5.25T + 29T^{2} \) |
| 31 | \( 1 - 0.391T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 + 2.56T + 43T^{2} \) |
| 47 | \( 1 + 0.994T + 47T^{2} \) |
| 53 | \( 1 - 0.241T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 7.98T + 67T^{2} \) |
| 71 | \( 1 + 2.49T + 71T^{2} \) |
| 73 | \( 1 - 1.44T + 73T^{2} \) |
| 79 | \( 1 + 9.05T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 2.19T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−7.69840664251508579393280312848, −7.14495182937311101673187487824, −6.87475951513251615231450879392, −5.52908798050670988898848069932, −5.08789789635618893252495040266, −4.70141504831143450069795305117, −4.14101310589537808731033201581, −2.90903748879546042969237238728, −1.77143447737463721297957899489, −0.05557492626907440603044800441,
0.05557492626907440603044800441, 1.77143447737463721297957899489, 2.90903748879546042969237238728, 4.14101310589537808731033201581, 4.70141504831143450069795305117, 5.08789789635618893252495040266, 5.52908798050670988898848069932, 6.87475951513251615231450879392, 7.14495182937311101673187487824, 7.69840664251508579393280312848
