| L(s) = 1 | + 3·3-s + 12.8·5-s − 24.8·7-s + 9·9-s − 19.6·11-s − 13·13-s + 38.4·15-s − 63.6·17-s + 0.832·19-s − 74.4·21-s − 119.·23-s + 39.6·25-s + 27·27-s − 6·29-s − 185.·31-s − 58.9·33-s − 318.·35-s + 143.·37-s − 39·39-s − 117.·41-s − 67.6·43-s + 115.·45-s − 476.·47-s + 273.·49-s − 190.·51-s + 59.3·53-s − 252.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.14·5-s − 1.34·7-s + 0.333·9-s − 0.539·11-s − 0.277·13-s + 0.662·15-s − 0.908·17-s + 0.0100·19-s − 0.774·21-s − 1.08·23-s + 0.317·25-s + 0.192·27-s − 0.0384·29-s − 1.07·31-s − 0.311·33-s − 1.53·35-s + 0.638·37-s − 0.160·39-s − 0.446·41-s − 0.240·43-s + 0.382·45-s − 1.47·47-s + 0.797·49-s − 0.524·51-s + 0.153·53-s − 0.618·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 - 12.8T + 125T^{2} \) |
| 7 | \( 1 + 24.8T + 343T^{2} \) |
| 11 | \( 1 + 19.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 63.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 0.832T + 6.85e3T^{2} \) |
| 23 | \( 1 + 119.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 143.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 117.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 67.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 476.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 59.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 78T + 2.05e5T^{2} \) |
| 61 | \( 1 - 609.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 654.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 390.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 84.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 430.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 802.T + 9.12e5T^{2} \) |
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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
−9.795812996104826114599932558923, −9.124363677545077858483171910646, −8.115312517894677901387090291526, −6.93892979440918666613253170259, −6.23426451183506329124542461925, −5.29127477641286509189773379133, −3.90839162036481011806053614130, −2.78028813104630432768662516022, −1.91664318989257877164704587627, 0,
1.91664318989257877164704587627, 2.78028813104630432768662516022, 3.90839162036481011806053614130, 5.29127477641286509189773379133, 6.23426451183506329124542461925, 6.93892979440918666613253170259, 8.115312517894677901387090291526, 9.124363677545077858483171910646, 9.795812996104826114599932558923
