| L(s) = 1 | + 1.71·3-s − 5-s + 2.39·7-s − 0.0460·9-s + 4.78·11-s + 4.50·13-s − 1.71·15-s + 0.391·17-s − 3.43·19-s + 4.11·21-s + 1.71·23-s + 25-s − 5.23·27-s − 29-s + 3.73·31-s + 8.22·33-s − 2.39·35-s − 2.78·37-s + 7.73·39-s + 6.78·41-s + 11.9·43-s + 0.0460·45-s + 8·47-s − 1.28·49-s + 0.672·51-s + 5.17·53-s − 4.78·55-s + ⋯ |
| L(s) = 1 | + 0.992·3-s − 0.447·5-s + 0.903·7-s − 0.0153·9-s + 1.44·11-s + 1.24·13-s − 0.443·15-s + 0.0949·17-s − 0.788·19-s + 0.896·21-s + 0.358·23-s + 0.200·25-s − 1.00·27-s − 0.185·29-s + 0.671·31-s + 1.43·33-s − 0.404·35-s − 0.457·37-s + 1.23·39-s + 1.05·41-s + 1.82·43-s + 0.00686·45-s + 1.16·47-s − 0.183·49-s + 0.0941·51-s + 0.710·53-s − 0.644·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.720493420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.720493420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 7 | \( 1 - 2.39T + 7T^{2} \) |
| 11 | \( 1 - 4.78T + 11T^{2} \) |
| 13 | \( 1 - 4.50T + 13T^{2} \) |
| 17 | \( 1 - 0.391T + 17T^{2} \) |
| 19 | \( 1 + 3.43T + 19T^{2} \) |
| 23 | \( 1 - 1.71T + 23T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 + 2.78T + 37T^{2} \) |
| 41 | \( 1 - 6.78T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 + 9.93T + 59T^{2} \) |
| 61 | \( 1 + 3.71T + 61T^{2} \) |
| 67 | \( 1 + 6.87T + 67T^{2} \) |
| 71 | \( 1 + 6.87T + 71T^{2} \) |
| 73 | \( 1 - 5.70T + 73T^{2} \) |
| 79 | \( 1 + 9.19T + 79T^{2} \) |
| 83 | \( 1 + 4.56T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 9.06T + 97T^{2} \) |
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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
−7.74126349950501462541562855200, −7.30987852720535711114732358667, −6.25142660006835021157595750352, −5.87621415904327773150768133943, −4.64921311946167547257563125512, −4.08514699226180729474388391921, −3.54608199143841908442437293087, −2.65717274648494185680226972208, −1.73990549438417072473097870768, −0.948241449822171624294540680858,
0.948241449822171624294540680858, 1.73990549438417072473097870768, 2.65717274648494185680226972208, 3.54608199143841908442437293087, 4.08514699226180729474388391921, 4.64921311946167547257563125512, 5.87621415904327773150768133943, 6.25142660006835021157595750352, 7.30987852720535711114732358667, 7.74126349950501462541562855200
