| L(s) = 1 | + 2.99·3-s − 5·5-s + 22.8·7-s − 18.0·9-s − 26.8·11-s − 29.4·13-s − 14.9·15-s − 40.2·17-s + 155.·19-s + 68.5·21-s + 23·23-s + 25·25-s − 134.·27-s − 101.·29-s + 167.·31-s − 80.4·33-s − 114.·35-s + 87.1·37-s − 88.1·39-s − 57.3·41-s + 434.·43-s + 90.1·45-s − 305.·47-s + 181.·49-s − 120.·51-s − 722.·53-s + 134.·55-s + ⋯ |
| L(s) = 1 | + 0.576·3-s − 0.447·5-s + 1.23·7-s − 0.667·9-s − 0.736·11-s − 0.627·13-s − 0.257·15-s − 0.574·17-s + 1.87·19-s + 0.712·21-s + 0.208·23-s + 0.200·25-s − 0.961·27-s − 0.651·29-s + 0.968·31-s − 0.424·33-s − 0.552·35-s + 0.387·37-s − 0.361·39-s − 0.218·41-s + 1.54·43-s + 0.298·45-s − 0.947·47-s + 0.528·49-s − 0.331·51-s − 1.87·53-s + 0.329·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 2.99T + 27T^{2} \) |
| 7 | \( 1 - 22.8T + 343T^{2} \) |
| 11 | \( 1 + 26.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 29.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 155.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 101.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 87.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 57.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 434.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 722.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 592.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 319.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 37.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 958.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 514.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 373.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 806.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.29e3T + 9.12e5T^{2} \) |
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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
−8.370069668629297493387459422148, −7.72650019177591451822841782200, −7.35747403763481101862794416640, −5.95652290500521543746732995031, −5.09529609659948523042757308725, −4.49178042702709683809108520225, −3.22492054905952247804863664688, −2.55011976734063389878485165184, −1.38093492662889465375602379225, 0,
1.38093492662889465375602379225, 2.55011976734063389878485165184, 3.22492054905952247804863664688, 4.49178042702709683809108520225, 5.09529609659948523042757308725, 5.95652290500521543746732995031, 7.35747403763481101862794416640, 7.72650019177591451822841782200, 8.370069668629297493387459422148
