| L(s) = 1 | − 2.92·3-s − 14.3·5-s + 16.8·7-s − 18.4·9-s − 16.5·11-s − 55.3·13-s + 41.9·15-s + 6.28·17-s + 119.·19-s − 49.3·21-s + 22.5·23-s + 80.3·25-s + 132.·27-s + 29·29-s + 228.·31-s + 48.4·33-s − 241.·35-s − 257.·37-s + 162.·39-s + 382.·41-s + 170.·43-s + 264.·45-s − 172.·47-s − 58.0·49-s − 18.3·51-s + 69.2·53-s + 237.·55-s + ⋯ |
| L(s) = 1 | − 0.563·3-s − 1.28·5-s + 0.911·7-s − 0.682·9-s − 0.453·11-s − 1.18·13-s + 0.721·15-s + 0.0896·17-s + 1.43·19-s − 0.513·21-s + 0.204·23-s + 0.643·25-s + 0.947·27-s + 0.185·29-s + 1.32·31-s + 0.255·33-s − 1.16·35-s − 1.14·37-s + 0.665·39-s + 1.45·41-s + 0.603·43-s + 0.875·45-s − 0.536·47-s − 0.169·49-s − 0.0504·51-s + 0.179·53-s + 0.581·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 2.92T + 27T^{2} \) |
| 5 | \( 1 + 14.3T + 125T^{2} \) |
| 7 | \( 1 - 16.8T + 343T^{2} \) |
| 11 | \( 1 + 16.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.28T + 4.91e3T^{2} \) |
| 19 | \( 1 - 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 22.5T + 1.21e4T^{2} \) |
| 31 | \( 1 - 228.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 382.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 170.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 172.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 69.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 43.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 684.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 528.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 488.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 80.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 741.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.36e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 957.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.11e3T + 9.12e5T^{2} \) |
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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
−8.245239835492174750396920760825, −7.72551887701432443367411095641, −7.15460395423462075077261018574, −5.96851108354577722901397323083, −5.00670269162153250844080281020, −4.66788954990448980818214904834, −3.41562742675031167577662345994, −2.52574442124650147195048515656, −0.972783394905675855366192136065, 0,
0.972783394905675855366192136065, 2.52574442124650147195048515656, 3.41562742675031167577662345994, 4.66788954990448980818214904834, 5.00670269162153250844080281020, 5.96851108354577722901397323083, 7.15460395423462075077261018574, 7.72551887701432443367411095641, 8.245239835492174750396920760825
