| L(s) = 1 | + 2·3-s + 5-s + 4.82·7-s + 9-s − 0.828·11-s − 2·13-s + 2·15-s + 2.82·17-s + 4.82·19-s + 9.65·21-s + 3.17·23-s + 25-s − 4·27-s + 29-s − 6.48·31-s − 1.65·33-s + 4.82·35-s − 8.48·37-s − 4·39-s − 6·41-s + 6·43-s + 45-s + 11.6·47-s + 16.3·49-s + 5.65·51-s − 3.65·53-s − 0.828·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1.82·7-s + 0.333·9-s − 0.249·11-s − 0.554·13-s + 0.516·15-s + 0.685·17-s + 1.10·19-s + 2.10·21-s + 0.661·23-s + 0.200·25-s − 0.769·27-s + 0.185·29-s − 1.16·31-s − 0.288·33-s + 0.816·35-s − 1.39·37-s − 0.640·39-s − 0.937·41-s + 0.914·43-s + 0.149·45-s + 1.70·47-s + 2.33·49-s + 0.792·51-s − 0.502·53-s − 0.111·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.572340825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.572340825\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2T + 3T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 + 7.17T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.962904947363668239949404092721, −8.187105038033051531234353676419, −7.66054007701108422732660305999, −7.04982665242014689501163186102, −5.41824376991538541362479969331, −5.25352859466767289010526269798, −4.06940339433377634459477286664, −3.05620024168407827640737649166, −2.17664833593556625923820737292, −1.33597335116244657555788678106,
1.33597335116244657555788678106, 2.17664833593556625923820737292, 3.05620024168407827640737649166, 4.06940339433377634459477286664, 5.25352859466767289010526269798, 5.41824376991538541362479969331, 7.04982665242014689501163186102, 7.66054007701108422732660305999, 8.187105038033051531234353676419, 8.962904947363668239949404092721
