L(s) = 1 | + 0.193·2-s + 2.50·3-s − 1.96·4-s − 5-s + 0.483·6-s − 0.166·7-s − 0.766·8-s + 3.25·9-s − 0.193·10-s − 0.120·11-s − 4.90·12-s + 4.23·13-s − 0.0320·14-s − 2.50·15-s + 3.77·16-s + 5.01·17-s + 0.629·18-s − 2.77·19-s + 1.96·20-s − 0.415·21-s − 0.0233·22-s + 5.34·23-s − 1.91·24-s + 25-s + 0.819·26-s + 0.645·27-s + 0.325·28-s + ⋯ |
L(s) = 1 | + 0.136·2-s + 1.44·3-s − 0.981·4-s − 0.447·5-s + 0.197·6-s − 0.0627·7-s − 0.270·8-s + 1.08·9-s − 0.0611·10-s − 0.0363·11-s − 1.41·12-s + 1.17·13-s − 0.00857·14-s − 0.645·15-s + 0.944·16-s + 1.21·17-s + 0.148·18-s − 0.637·19-s + 0.438·20-s − 0.0906·21-s − 0.00496·22-s + 1.11·23-s − 0.391·24-s + 0.200·25-s + 0.160·26-s + 0.124·27-s + 0.0615·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.269584017\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.269584017\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 0.193T + 2T^{2} \) |
| 3 | \( 1 - 2.50T + 3T^{2} \) |
| 7 | \( 1 + 0.166T + 7T^{2} \) |
| 11 | \( 1 + 0.120T + 11T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 + 2.77T + 19T^{2} \) |
| 23 | \( 1 - 5.34T + 23T^{2} \) |
| 29 | \( 1 - 0.290T + 29T^{2} \) |
| 31 | \( 1 - 3.57T + 31T^{2} \) |
| 37 | \( 1 - 0.104T + 37T^{2} \) |
| 41 | \( 1 - 7.23T + 41T^{2} \) |
| 43 | \( 1 - 6.76T + 43T^{2} \) |
| 47 | \( 1 - 1.90T + 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 + 6.34T + 59T^{2} \) |
| 61 | \( 1 + 0.0566T + 61T^{2} \) |
| 67 | \( 1 - 6.47T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 4.84T + 83T^{2} \) |
| 89 | \( 1 + 5.92T + 89T^{2} \) |
| 97 | \( 1 + 4.77T + 97T^{2} \) |
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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
−9.350042236780484386882663736758, −8.987908295209262225020341413717, −8.082790243506617561404666954372, −7.80284426431120113469814361716, −6.46047386255506990092088708000, −5.34801575000518203287049967003, −4.22898770202257700334109315031, −3.58217787749460642750743540308, −2.79769996168839786682085646919, −1.12911514224724749869524462489,
1.12911514224724749869524462489, 2.79769996168839786682085646919, 3.58217787749460642750743540308, 4.22898770202257700334109315031, 5.34801575000518203287049967003, 6.46047386255506990092088708000, 7.80284426431120113469814361716, 8.082790243506617561404666954372, 8.987908295209262225020341413717, 9.350042236780484386882663736758