| L(s) = 1 | + 3-s − 2.82·5-s − 7-s + 9-s − 2·11-s + 2.82·13-s − 2.82·15-s − 19-s − 21-s + 0.828·23-s + 3.00·25-s + 27-s + 7.65·29-s − 6.82·31-s − 2·33-s + 2.82·35-s + 10.4·37-s + 2.82·39-s + 3.65·41-s − 8·43-s − 2.82·45-s + 2.82·47-s + 49-s − 3.65·53-s + 5.65·55-s − 57-s + 9.65·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.26·5-s − 0.377·7-s + 0.333·9-s − 0.603·11-s + 0.784·13-s − 0.730·15-s − 0.229·19-s − 0.218·21-s + 0.172·23-s + 0.600·25-s + 0.192·27-s + 1.42·29-s − 1.22·31-s − 0.348·33-s + 0.478·35-s + 1.72·37-s + 0.452·39-s + 0.571·41-s − 1.21·43-s − 0.421·45-s + 0.412·47-s + 0.142·49-s − 0.502·53-s + 0.762·55-s − 0.132·57-s + 1.25·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 9.65T + 71T^{2} \) |
| 73 | \( 1 + 0.343T + 73T^{2} \) |
| 79 | \( 1 - 3.17T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.86936009335166280823836277351, −7.10168349987249283410269549321, −6.41345368500381701368320915916, −5.52939615381435575439154500035, −4.52317245590552112360044059549, −3.98371667735616950321781318018, −3.22037150882976994374167045890, −2.56069661635253052208610395959, −1.23156185915157281683590494997, 0,
1.23156185915157281683590494997, 2.56069661635253052208610395959, 3.22037150882976994374167045890, 3.98371667735616950321781318018, 4.52317245590552112360044059549, 5.52939615381435575439154500035, 6.41345368500381701368320915916, 7.10168349987249283410269549321, 7.86936009335166280823836277351
