| L(s) = 1 | − 3·3-s + 12.8·5-s + 9·9-s + 36.8·11-s − 87.1·13-s − 38.4·15-s − 102.·17-s + 95.8·19-s + 96·23-s + 39.4·25-s − 27·27-s − 212.·29-s − 159.·31-s − 110.·33-s + 128.·37-s + 261.·39-s + 298.·41-s + 33.3·43-s + 115.·45-s + 271.·47-s + 307.·51-s + 448.·53-s + 472.·55-s − 287.·57-s − 668.·59-s + 243.·61-s − 1.11e3·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.14·5-s + 0.333·9-s + 1.00·11-s − 1.85·13-s − 0.662·15-s − 1.46·17-s + 1.15·19-s + 0.870·23-s + 0.315·25-s − 0.192·27-s − 1.35·29-s − 0.922·31-s − 0.582·33-s + 0.571·37-s + 1.07·39-s + 1.13·41-s + 0.118·43-s + 0.382·45-s + 0.841·47-s + 0.845·51-s + 1.16·53-s + 1.15·55-s − 0.668·57-s − 1.47·59-s + 0.511·61-s − 2.13·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.022440010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022440010\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 12.8T + 125T^{2} \) |
| 11 | \( 1 - 36.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 87.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 95.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 96T + 1.21e4T^{2} \) |
| 29 | \( 1 + 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 128.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 33.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 271.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 448.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 668.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 243.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 335.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 339.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 918.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 136.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 287.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 161.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 182.T + 9.12e5T^{2} \) |
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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
−9.103464380242617553511098400996, −7.54629153979818213215710048868, −7.06807268117595668821653741597, −6.25995647612891808886014689614, −5.47227013401897378005324302583, −4.86869187155447961250320973521, −3.90041283273099290310489182724, −2.54251591136882356422614091910, −1.84786725164573759329089512512, −0.64608099154644419861153798470,
0.64608099154644419861153798470, 1.84786725164573759329089512512, 2.54251591136882356422614091910, 3.90041283273099290310489182724, 4.86869187155447961250320973521, 5.47227013401897378005324302583, 6.25995647612891808886014689614, 7.06807268117595668821653741597, 7.54629153979818213215710048868, 9.103464380242617553511098400996
