L(s) = 1 | − 2.25·3-s + 3.53·5-s + 1.19·7-s + 2.06·9-s − 4.93·11-s − 4.24·13-s − 7.95·15-s − 0.933·17-s + 5.44·19-s − 2.67·21-s − 8.71·23-s + 7.47·25-s + 2.10·27-s + 8.13·29-s + 5.52·31-s + 11.1·33-s + 4.20·35-s + 3.19·37-s + 9.54·39-s + 11.8·41-s − 9.73·43-s + 7.30·45-s − 10.8·47-s − 5.58·49-s + 2.10·51-s − 3.26·53-s − 17.4·55-s + ⋯ |
L(s) = 1 | − 1.29·3-s + 1.57·5-s + 0.449·7-s + 0.688·9-s − 1.48·11-s − 1.17·13-s − 2.05·15-s − 0.226·17-s + 1.24·19-s − 0.584·21-s − 1.81·23-s + 1.49·25-s + 0.404·27-s + 1.51·29-s + 0.992·31-s + 1.93·33-s + 0.710·35-s + 0.525·37-s + 1.52·39-s + 1.85·41-s − 1.48·43-s + 1.08·45-s − 1.58·47-s − 0.797·49-s + 0.294·51-s − 0.449·53-s − 2.35·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 2.25T + 3T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 4.93T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 0.933T + 17T^{2} \) |
| 19 | \( 1 - 5.44T + 19T^{2} \) |
| 23 | \( 1 + 8.71T + 23T^{2} \) |
| 29 | \( 1 - 8.13T + 29T^{2} \) |
| 31 | \( 1 - 5.52T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 9.73T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 3.26T + 53T^{2} \) |
| 59 | \( 1 - 1.18T + 59T^{2} \) |
| 61 | \( 1 + 1.59T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 - 7.27T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 2.87T + 89T^{2} \) |
| 97 | \( 1 + 4.60T + 97T^{2} \) |
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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
−7.79058227282698365898396863483, −6.69320213273672289466544124874, −6.20628869163903044647115674975, −5.50641857980538578524181483151, −5.04338976177392749783212779477, −4.58950213684916665776419000204, −2.87730307117529492364632671193, −2.31179363370600287484719759769, −1.25423428595900978735453049610, 0,
1.25423428595900978735453049610, 2.31179363370600287484719759769, 2.87730307117529492364632671193, 4.58950213684916665776419000204, 5.04338976177392749783212779477, 5.50641857980538578524181483151, 6.20628869163903044647115674975, 6.69320213273672289466544124874, 7.79058227282698365898396863483