| L(s) = 1 | − 5.59·3-s + 20.9·5-s + 13.9·7-s + 4.32·9-s + 58.3·11-s − 65.0·13-s − 117.·15-s + 31.7·17-s + 19·19-s − 77.8·21-s + 151.·23-s + 314.·25-s + 126.·27-s − 110.·29-s + 94.0·31-s − 326.·33-s + 291.·35-s + 291.·37-s + 364.·39-s + 64.4·41-s − 449.·43-s + 90.7·45-s + 530.·47-s − 149.·49-s − 177.·51-s − 621.·53-s + 1.22e3·55-s + ⋯ |
| L(s) = 1 | − 1.07·3-s + 1.87·5-s + 0.750·7-s + 0.160·9-s + 1.59·11-s − 1.38·13-s − 2.01·15-s + 0.452·17-s + 0.229·19-s − 0.808·21-s + 1.37·23-s + 2.51·25-s + 0.904·27-s − 0.710·29-s + 0.544·31-s − 1.72·33-s + 1.40·35-s + 1.29·37-s + 1.49·39-s + 0.245·41-s − 1.59·43-s + 0.300·45-s + 1.64·47-s − 0.436·49-s − 0.487·51-s − 1.60·53-s + 2.99·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.599335106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.599335106\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 5.59T + 27T^{2} \) |
| 5 | \( 1 - 20.9T + 125T^{2} \) |
| 7 | \( 1 - 13.9T + 343T^{2} \) |
| 11 | \( 1 - 58.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.7T + 4.91e3T^{2} \) |
| 23 | \( 1 - 151.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 94.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 291.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 64.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 449.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 530.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 621.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 244.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 801.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 7.34T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 592.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 120.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 502.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 253.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 511.T + 9.12e5T^{2} \) |
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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
−9.441351311725325817421782780733, −8.894389271212452261446857546170, −7.47727775474787294188713377139, −6.55083912726725759546278938333, −6.02786992957569520963985683754, −5.14927401537381944682082273470, −4.68122339179542285436123516873, −2.91357626661144613290441531327, −1.75218045256932573854477551237, −0.943845001286390821224885017792,
0.943845001286390821224885017792, 1.75218045256932573854477551237, 2.91357626661144613290441531327, 4.68122339179542285436123516873, 5.14927401537381944682082273470, 6.02786992957569520963985683754, 6.55083912726725759546278938333, 7.47727775474787294188713377139, 8.894389271212452261446857546170, 9.441351311725325817421782780733
