L(s) = 1 | − 2.10·2-s − 0.209·3-s + 2.42·4-s + 1.85·5-s + 0.441·6-s + 3.06·7-s − 0.900·8-s − 2.95·9-s − 3.91·10-s − 11-s − 0.508·12-s + 0.726·13-s − 6.44·14-s − 0.389·15-s − 2.96·16-s + 17-s + 6.22·18-s + 5.08·19-s + 4.51·20-s − 0.641·21-s + 2.10·22-s − 7.62·23-s + 0.188·24-s − 1.54·25-s − 1.52·26-s + 1.24·27-s + 7.43·28-s + ⋯ |
L(s) = 1 | − 1.48·2-s − 0.121·3-s + 1.21·4-s + 0.831·5-s + 0.180·6-s + 1.15·7-s − 0.318·8-s − 0.985·9-s − 1.23·10-s − 0.301·11-s − 0.146·12-s + 0.201·13-s − 1.72·14-s − 0.100·15-s − 0.740·16-s + 0.242·17-s + 1.46·18-s + 1.16·19-s + 1.00·20-s − 0.140·21-s + 0.448·22-s − 1.59·23-s + 0.0385·24-s − 0.308·25-s − 0.299·26-s + 0.240·27-s + 1.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 3 | \( 1 + 0.209T + 3T^{2} \) |
| 5 | \( 1 - 1.85T + 5T^{2} \) |
| 7 | \( 1 - 3.06T + 7T^{2} \) |
| 13 | \( 1 - 0.726T + 13T^{2} \) |
| 19 | \( 1 - 5.08T + 19T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 - 0.475T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 - 9.14T + 41T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 - 9.59T + 53T^{2} \) |
| 59 | \( 1 + 4.55T + 59T^{2} \) |
| 61 | \( 1 - 2.60T + 61T^{2} \) |
| 67 | \( 1 + 0.886T + 67T^{2} \) |
| 71 | \( 1 + 7.16T + 71T^{2} \) |
| 73 | \( 1 - 1.47T + 73T^{2} \) |
| 79 | \( 1 - 1.82T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 - 2.12T + 89T^{2} \) |
| 97 | \( 1 + 9.30T + 97T^{2} \) |
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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
−7.66959280809752635405621031559, −7.18895465783325755926567172743, −5.96869502844439970501913037770, −5.69221569725870116700735704299, −4.87521869344566415950717924578, −3.82872412606421207899498818485, −2.61591064489609483588242896179, −1.92928354683588188024096005839, −1.22638188940017873622092749640, 0,
1.22638188940017873622092749640, 1.92928354683588188024096005839, 2.61591064489609483588242896179, 3.82872412606421207899498818485, 4.87521869344566415950717924578, 5.69221569725870116700735704299, 5.96869502844439970501913037770, 7.18895465783325755926567172743, 7.66959280809752635405621031559