| L(s) = 1 | + 2.10·3-s − 3.40·5-s + 7-s + 1.43·9-s + 1.23·11-s − 6.47·13-s − 7.17·15-s + 17-s + 2.21·19-s + 2.10·21-s − 1.39·23-s + 6.60·25-s − 3.30·27-s − 0.633·29-s − 0.965·31-s + 2.60·33-s − 3.40·35-s − 8.31·37-s − 13.6·39-s − 5.60·41-s − 11.0·43-s − 4.87·45-s − 7.67·47-s + 49-s + 2.10·51-s + 11.7·53-s − 4.21·55-s + ⋯ |
| L(s) = 1 | + 1.21·3-s − 1.52·5-s + 0.377·7-s + 0.477·9-s + 0.372·11-s − 1.79·13-s − 1.85·15-s + 0.242·17-s + 0.507·19-s + 0.459·21-s − 0.291·23-s + 1.32·25-s − 0.635·27-s − 0.117·29-s − 0.173·31-s + 0.452·33-s − 0.575·35-s − 1.36·37-s − 2.18·39-s − 0.875·41-s − 1.68·43-s − 0.727·45-s − 1.11·47-s + 0.142·49-s + 0.294·51-s + 1.60·53-s − 0.567·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.10T + 3T^{2} \) |
| 5 | \( 1 + 3.40T + 5T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 + 1.39T + 23T^{2} \) |
| 29 | \( 1 + 0.633T + 29T^{2} \) |
| 31 | \( 1 + 0.965T + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 7.67T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 0.602T + 59T^{2} \) |
| 61 | \( 1 + 9.84T + 61T^{2} \) |
| 67 | \( 1 + 5.30T + 67T^{2} \) |
| 71 | \( 1 - 3.33T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 - 0.323T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 5.90T + 97T^{2} \) |
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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
−8.529725784747750142335040694478, −8.163368493990852981866134151074, −7.36828435050252635395689842704, −6.96533782031094690438592464026, −5.32454458171692868630858501985, −4.53558909667300863919179729368, −3.62498850997454474272934263897, −3.00859459784992687899187139590, −1.85430439698695992713526001106, 0,
1.85430439698695992713526001106, 3.00859459784992687899187139590, 3.62498850997454474272934263897, 4.53558909667300863919179729368, 5.32454458171692868630858501985, 6.96533782031094690438592464026, 7.36828435050252635395689842704, 8.163368493990852981866134151074, 8.529725784747750142335040694478
