| L(s) = 1 | + 9.09·3-s − 5·5-s − 6.08·7-s + 55.6·9-s − 23.3·11-s + 37.3·13-s − 45.4·15-s − 91.4·17-s + 4.18·19-s − 55.3·21-s − 23·23-s + 25·25-s + 260.·27-s − 251.·29-s − 305.·31-s − 212.·33-s + 30.4·35-s + 262.·37-s + 339.·39-s − 288.·41-s + 377.·43-s − 278.·45-s − 463.·47-s − 305.·49-s − 831.·51-s − 581.·53-s + 116.·55-s + ⋯ |
| L(s) = 1 | + 1.74·3-s − 0.447·5-s − 0.328·7-s + 2.06·9-s − 0.641·11-s + 0.795·13-s − 0.782·15-s − 1.30·17-s + 0.0505·19-s − 0.575·21-s − 0.208·23-s + 0.200·25-s + 1.85·27-s − 1.60·29-s − 1.77·31-s − 1.12·33-s + 0.147·35-s + 1.16·37-s + 1.39·39-s − 1.10·41-s + 1.33·43-s − 0.922·45-s − 1.43·47-s − 0.891·49-s − 2.28·51-s − 1.50·53-s + 0.286·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 9.09T + 27T^{2} \) |
| 7 | \( 1 + 6.08T + 343T^{2} \) |
| 11 | \( 1 + 23.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 91.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.18T + 6.85e3T^{2} \) |
| 29 | \( 1 + 251.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 305.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 262.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 288.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 581.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 58.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 45.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 772.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 16.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 664.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 187.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 730.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 212.T + 9.12e5T^{2} \) |
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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
−8.399299816516183361196296724248, −7.927218512807174153064931114343, −7.18914010769166381173708703186, −6.31972711134057051895582259280, −5.03262362698262697409242249304, −3.93223586483696102654833428881, −3.48610575703025568205920044183, −2.47852060462931263075991599772, −1.67028228349539981585991990580, 0,
1.67028228349539981585991990580, 2.47852060462931263075991599772, 3.48610575703025568205920044183, 3.93223586483696102654833428881, 5.03262362698262697409242249304, 6.31972711134057051895582259280, 7.18914010769166381173708703186, 7.927218512807174153064931114343, 8.399299816516183361196296724248
