| L(s) = 1 | + 3-s + 0.703·5-s − 7-s + 9-s − 0.606·11-s − 1.30·13-s + 0.703·15-s − 3.67·17-s + 0.606·19-s − 21-s + 23-s − 4.50·25-s + 27-s + 5.63·29-s − 1.86·31-s − 0.606·33-s − 0.703·35-s − 2.80·37-s − 1.30·39-s − 0.329·41-s + 2.99·43-s + 0.703·45-s + 4.19·47-s + 49-s − 3.67·51-s − 10.0·53-s − 0.426·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.314·5-s − 0.377·7-s + 0.333·9-s − 0.182·11-s − 0.363·13-s + 0.181·15-s − 0.890·17-s + 0.139·19-s − 0.218·21-s + 0.208·23-s − 0.900·25-s + 0.192·27-s + 1.04·29-s − 0.335·31-s − 0.105·33-s − 0.118·35-s − 0.460·37-s − 0.209·39-s − 0.0513·41-s + 0.456·43-s + 0.104·45-s + 0.611·47-s + 0.142·49-s − 0.514·51-s − 1.37·53-s − 0.0575·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 0.703T + 5T^{2} \) |
| 11 | \( 1 + 0.606T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 3.67T + 17T^{2} \) |
| 19 | \( 1 - 0.606T + 19T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 + 2.80T + 37T^{2} \) |
| 41 | \( 1 + 0.329T + 41T^{2} \) |
| 43 | \( 1 - 2.99T + 43T^{2} \) |
| 47 | \( 1 - 4.19T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 9.73T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 8.66T + 67T^{2} \) |
| 71 | \( 1 - 3.86T + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 - 1.86T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 5.82T + 97T^{2} \) |
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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
−7.53507968815591016922365090813, −6.85941438649516440666965323919, −6.22032836045820273334671888339, −5.42268859005913717433808459723, −4.60221215370727771804884439155, −3.91143905421051222283407749834, −2.97593440308285178608419789218, −2.37065408272788500202491933588, −1.42337354704132020736888631440, 0,
1.42337354704132020736888631440, 2.37065408272788500202491933588, 2.97593440308285178608419789218, 3.91143905421051222283407749834, 4.60221215370727771804884439155, 5.42268859005913717433808459723, 6.22032836045820273334671888339, 6.85941438649516440666965323919, 7.53507968815591016922365090813
