L(s) = 1 | − 0.618·2-s + 3.23·3-s − 1.61·4-s − 2.00·6-s + 2.23·8-s + 7.47·9-s − 0.236·11-s − 5.23·12-s − 1.23·13-s + 1.85·16-s − 2.47·17-s − 4.61·18-s + 4.47·19-s + 0.145·22-s + 6.23·23-s + 7.23·24-s + 0.763·26-s + 14.4·27-s + 5·29-s − 3.70·31-s − 5.61·32-s − 0.763·33-s + 1.52·34-s − 12.0·36-s + 3·37-s − 2.76·38-s − 4.00·39-s + ⋯ |
L(s) = 1 | − 0.437·2-s + 1.86·3-s − 0.809·4-s − 0.816·6-s + 0.790·8-s + 2.49·9-s − 0.0711·11-s − 1.51·12-s − 0.342·13-s + 0.463·16-s − 0.599·17-s − 1.08·18-s + 1.02·19-s + 0.0311·22-s + 1.30·23-s + 1.47·24-s + 0.149·26-s + 2.78·27-s + 0.928·29-s − 0.666·31-s − 0.993·32-s − 0.132·33-s + 0.262·34-s − 2.01·36-s + 0.493·37-s − 0.448·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.233891138\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.233891138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 3 | \( 1 - 3.23T + 3T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 9.70T + 61T^{2} \) |
| 67 | \( 1 + 4.23T + 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 - 8.76T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 + 5.23T + 97T^{2} \) |
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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
−9.493363259432632079369868474619, −8.890918003524942691538104282779, −8.327507700963415256516021348043, −7.53881528940075562291240651760, −6.91584616837430072010938452916, −5.19643903778862967809830211199, −4.34548315632484463919046543122, −3.43933625865169725969500505618, −2.52401877585253848364795705257, −1.21924125463355925731094325499,
1.21924125463355925731094325499, 2.52401877585253848364795705257, 3.43933625865169725969500505618, 4.34548315632484463919046543122, 5.19643903778862967809830211199, 6.91584616837430072010938452916, 7.53881528940075562291240651760, 8.327507700963415256516021348043, 8.890918003524942691538104282779, 9.493363259432632079369868474619