| L(s) = 1 | + 3·3-s + 18.8·7-s + 9·9-s + 63.2·11-s − 1.58·13-s − 135.·17-s − 97.2·19-s + 56.4·21-s + 41.1·23-s + 27·27-s − 207.·29-s − 193.·31-s + 189.·33-s − 339.·37-s − 4.74·39-s − 490.·41-s + 74.3·43-s + 544.·47-s + 10.6·49-s − 405.·51-s − 663.·53-s − 291.·57-s − 344.·59-s + 5.16·61-s + 169.·63-s − 671.·67-s + 123.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.01·7-s + 0.333·9-s + 1.73·11-s − 0.0337·13-s − 1.92·17-s − 1.17·19-s + 0.586·21-s + 0.373·23-s + 0.192·27-s − 1.32·29-s − 1.12·31-s + 1.00·33-s − 1.50·37-s − 0.0194·39-s − 1.86·41-s + 0.263·43-s + 1.68·47-s + 0.0311·49-s − 1.11·51-s − 1.72·53-s − 0.677·57-s − 0.760·59-s + 0.0108·61-s + 0.338·63-s − 1.22·67-s + 0.215·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 18.8T + 343T^{2} \) |
| 11 | \( 1 - 63.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 1.58T + 2.19e3T^{2} \) |
| 17 | \( 1 + 135.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 97.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 41.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 207.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 339.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 490.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 74.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 544.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 663.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 344.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 5.16T + 2.26e5T^{2} \) |
| 67 | \( 1 + 671.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 425.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 94.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 770.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 589.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 409.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 152.T + 9.12e5T^{2} \) |
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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
−8.567249803989425016264602492192, −7.37308043754691896300313953491, −6.83515254023079964037531547104, −6.00759698749254995906326416257, −4.78671852703368905592254667224, −4.22207232938182282814061260619, −3.42000321713912323497847140229, −1.95130555122531036253230939380, −1.66016345870819274787541508717, 0,
1.66016345870819274787541508717, 1.95130555122531036253230939380, 3.42000321713912323497847140229, 4.22207232938182282814061260619, 4.78671852703368905592254667224, 6.00759698749254995906326416257, 6.83515254023079964037531547104, 7.37308043754691896300313953491, 8.567249803989425016264602492192
