| L(s) = 1 | − 2.92·2-s + 4.54·3-s + 0.545·4-s + 6.17·5-s − 13.2·6-s − 32.1·7-s + 21.7·8-s − 6.37·9-s − 18.0·10-s + 2.47·12-s + 4.49·13-s + 93.9·14-s + 28.0·15-s − 68.0·16-s − 59.4·17-s + 18.6·18-s − 28.9·19-s + 3.37·20-s − 145.·21-s − 38.3·23-s + 98.9·24-s − 86.8·25-s − 13.1·26-s − 151.·27-s − 17.5·28-s − 39.4·29-s − 82.0·30-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.873·3-s + 0.0682·4-s + 0.552·5-s − 0.903·6-s − 1.73·7-s + 0.963·8-s − 0.236·9-s − 0.571·10-s + 0.0596·12-s + 0.0959·13-s + 1.79·14-s + 0.482·15-s − 1.06·16-s − 0.848·17-s + 0.244·18-s − 0.350·19-s + 0.0377·20-s − 1.51·21-s − 0.347·23-s + 0.841·24-s − 0.694·25-s − 0.0991·26-s − 1.08·27-s − 0.118·28-s − 0.252·29-s − 0.499·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| good | 2 | \( 1 + 2.92T + 8T^{2} \) |
| 3 | \( 1 - 4.54T + 27T^{2} \) |
| 5 | \( 1 - 6.17T + 125T^{2} \) |
| 7 | \( 1 + 32.1T + 343T^{2} \) |
| 13 | \( 1 - 4.49T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 38.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 39.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 266.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 112.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 134.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 182.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 42.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 180.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 559.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 770.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 26.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 372.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 252.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 58.5T + 7.04e5T^{2} \) |
| 97 | \( 1 + 597.T + 9.12e5T^{2} \) |
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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
−12.84278517096132338028130847715, −11.02838986345613117165335373952, −9.803641200028207352657294860959, −9.351614261943575474430850980476, −8.501513405801911020260283844566, −7.21806345298375201686282865924, −5.94370199404274316885814880068, −3.80881748214277372829553739273, −2.29173548197886473772154199433, 0,
2.29173548197886473772154199433, 3.80881748214277372829553739273, 5.94370199404274316885814880068, 7.21806345298375201686282865924, 8.501513405801911020260283844566, 9.351614261943575474430850980476, 9.803641200028207352657294860959, 11.02838986345613117165335373952, 12.84278517096132338028130847715
