L(s) = 1 | − 1.38·2-s − 1.64·3-s − 0.0828·4-s + 2.27·6-s − 2.06·7-s + 2.88·8-s − 0.288·9-s − 1.34·11-s + 0.136·12-s + 0.417·13-s + 2.85·14-s − 3.82·16-s − 7.21·17-s + 0.399·18-s − 4.92·19-s + 3.39·21-s + 1.85·22-s + 7.81·23-s − 4.74·24-s − 0.578·26-s + 5.41·27-s + 0.170·28-s − 5.56·29-s − 6.74·31-s − 0.468·32-s + 2.20·33-s + 9.99·34-s + ⋯ |
L(s) = 1 | − 0.979·2-s − 0.950·3-s − 0.0414·4-s + 0.930·6-s − 0.779·7-s + 1.01·8-s − 0.0962·9-s − 0.404·11-s + 0.0393·12-s + 0.115·13-s + 0.763·14-s − 0.956·16-s − 1.75·17-s + 0.0942·18-s − 1.12·19-s + 0.740·21-s + 0.395·22-s + 1.62·23-s − 0.969·24-s − 0.113·26-s + 1.04·27-s + 0.0322·28-s − 1.03·29-s − 1.21·31-s − 0.0827·32-s + 0.384·33-s + 1.71·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.01344457729\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01344457729\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 1.38T + 2T^{2} \) |
| 3 | \( 1 + 1.64T + 3T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 11 | \( 1 + 1.34T + 11T^{2} \) |
| 13 | \( 1 - 0.417T + 13T^{2} \) |
| 17 | \( 1 + 7.21T + 17T^{2} \) |
| 19 | \( 1 + 4.92T + 19T^{2} \) |
| 23 | \( 1 - 7.81T + 23T^{2} \) |
| 29 | \( 1 + 5.56T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 - 8.08T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 8.61T + 43T^{2} \) |
| 47 | \( 1 + 9.63T + 47T^{2} \) |
| 53 | \( 1 - 6.10T + 53T^{2} \) |
| 59 | \( 1 - 5.97T + 59T^{2} \) |
| 61 | \( 1 - 4.14T + 61T^{2} \) |
| 67 | \( 1 + 6.95T + 67T^{2} \) |
| 71 | \( 1 + 7.17T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 - 1.93T + 83T^{2} \) |
| 89 | \( 1 + 4.50T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−8.420382607084459822848553172277, −7.21011581339820345247167787880, −6.78955962968880734681435796309, −6.12159655204364137572510181079, −5.16875322697357583528874984975, −4.66354055622118080231044802784, −3.68666447757731128943998947165, −2.58082422408542266968985771596, −1.52117407751635688201705691378, −0.07499689058893673073988574290,
0.07499689058893673073988574290, 1.52117407751635688201705691378, 2.58082422408542266968985771596, 3.68666447757731128943998947165, 4.66354055622118080231044802784, 5.16875322697357583528874984975, 6.12159655204364137572510181079, 6.78955962968880734681435796309, 7.21011581339820345247167787880, 8.420382607084459822848553172277