L(s) = 1 | − 3-s + 3.85·5-s − 7-s + 9-s − 3·11-s − 0.381·13-s − 3.85·15-s − 1.47·17-s − 0.236·19-s + 21-s + 23-s + 9.85·25-s − 27-s − 5·29-s − 5.76·31-s + 3·33-s − 3.85·35-s − 4.23·37-s + 0.381·39-s + 3·41-s + 1.61·43-s + 3.85·45-s + 3.23·47-s + 49-s + 1.47·51-s − 2.32·53-s − 11.5·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.72·5-s − 0.377·7-s + 0.333·9-s − 0.904·11-s − 0.105·13-s − 0.995·15-s − 0.357·17-s − 0.0541·19-s + 0.218·21-s + 0.208·23-s + 1.97·25-s − 0.192·27-s − 0.928·29-s − 1.03·31-s + 0.522·33-s − 0.651·35-s − 0.696·37-s + 0.0611·39-s + 0.468·41-s + 0.246·43-s + 0.574·45-s + 0.472·47-s + 0.142·49-s + 0.206·51-s − 0.319·53-s − 1.55·55-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 | \( 1 - 3.85T + 5T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 0.381T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 + 0.236T + 19T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 5.76T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 - 3.23T + 47T^{2} \) |
| 53 | \( 1 + 2.32T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 4.85T + 61T^{2} \) |
| 67 | \( 1 + 2.32T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 + 5.94T + 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 6.56T + 89T^{2} \) |
| 97 | \( 1 + 1.94T + 97T^{2} \) |
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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
−7.23478982035523357998128040699, −6.80341562784147950755715584878, −5.88370370992238350283274031557, −5.59512852094881163742342695829, −4.99405903155689765441514293513, −4.00453702296554833361948796494, −2.88228964439667777721516428351, −2.20063547596711999867562317518, −1.37767614086931595311902884736, 0,
1.37767614086931595311902884736, 2.20063547596711999867562317518, 2.88228964439667777721516428351, 4.00453702296554833361948796494, 4.99405903155689765441514293513, 5.59512852094881163742342695829, 5.88370370992238350283274031557, 6.80341562784147950755715584878, 7.23478982035523357998128040699