| L(s) = 1 | + 3·3-s + 5·5-s − 9.69·7-s + 9·9-s − 19.6·11-s − 41.2·13-s + 15·15-s − 29.0·17-s + 140.·19-s − 29.0·21-s − 23·23-s + 25·25-s + 27·27-s + 54.8·29-s − 37.5·31-s − 59.0·33-s − 48.4·35-s − 184.·37-s − 123.·39-s + 509.·41-s − 478.·43-s + 45·45-s − 171.·47-s − 248.·49-s − 87.1·51-s − 374.·53-s − 98.4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.523·7-s + 0.333·9-s − 0.539·11-s − 0.879·13-s + 0.258·15-s − 0.414·17-s + 1.69·19-s − 0.302·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.351·29-s − 0.217·31-s − 0.311·33-s − 0.234·35-s − 0.821·37-s − 0.507·39-s + 1.93·41-s − 1.69·43-s + 0.149·45-s − 0.532·47-s − 0.725·49-s − 0.239·51-s − 0.970·53-s − 0.241·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 7 | \( 1 + 9.69T + 343T^{2} \) |
| 11 | \( 1 + 19.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 29.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 140.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 54.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 37.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 184.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 509.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 478.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 171.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 374.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 725.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 118.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 47.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 341.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 814.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 738.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 693.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 39.2T + 9.12e5T^{2} \) |
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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
−8.950537740925223960675038008961, −7.922549667154615921612669757054, −7.29604583706322528761343637707, −6.40929518616401076769387393241, −5.39768783025361711822716963429, −4.61557512370807013849246498535, −3.31487678338623730839888935976, −2.66900940567095443731064340539, −1.50797511107547332303009451783, 0,
1.50797511107547332303009451783, 2.66900940567095443731064340539, 3.31487678338623730839888935976, 4.61557512370807013849246498535, 5.39768783025361711822716963429, 6.40929518616401076769387393241, 7.29604583706322528761343637707, 7.922549667154615921612669757054, 8.950537740925223960675038008961
