| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s + 6·13-s − 2·15-s + 2·17-s + 8·19-s + 21-s − 4·23-s − 25-s − 27-s − 2·29-s − 8·31-s − 2·35-s + 6·37-s − 6·39-s + 2·41-s − 8·43-s + 2·45-s − 4·47-s + 49-s − 2·51-s + 2·53-s − 8·57-s − 12·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.516·15-s + 0.485·17-s + 1.83·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.338·35-s + 0.986·37-s − 0.960·39-s + 0.312·41-s − 1.21·43-s + 0.298·45-s − 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 1.05·57-s − 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.10155711634762, −15.49067045346787, −14.84333209081569, −14.07258015537726, −13.69049912400547, −13.25113043584245, −12.75848114128607, −11.94014391308877, −11.56162371174715, −10.97458075503436, −10.28043301297099, −9.918801314336757, −9.199844110011876, −8.912192039993417, −7.744580967618532, −7.581566935799113, −6.545172276310187, −6.026794729637040, −5.720752487062313, −5.107058953326645, −4.147285493358232, −3.481707232160470, −2.867647858764529, −1.589032499342947, −1.340149940548437, 0,
1.340149940548437, 1.589032499342947, 2.867647858764529, 3.481707232160470, 4.147285493358232, 5.107058953326645, 5.720752487062313, 6.026794729637040, 6.545172276310187, 7.581566935799113, 7.744580967618532, 8.912192039993417, 9.199844110011876, 9.918801314336757, 10.28043301297099, 10.97458075503436, 11.56162371174715, 11.94014391308877, 12.75848114128607, 13.25113043584245, 13.69049912400547, 14.07258015537726, 14.84333209081569, 15.49067045346787, 16.10155711634762
