| L(s) = 1 | + 3-s − 5-s − 3.23·7-s + 9-s + 1.23·13-s − 15-s − 4.47·17-s + 2.47·19-s − 3.23·21-s + 4·23-s + 25-s + 27-s − 2.76·29-s + 31-s + 3.23·35-s + 9.23·37-s + 1.23·39-s − 6·41-s + 2.47·43-s − 45-s + 8.94·47-s + 3.47·49-s − 4.47·51-s − 0.472·53-s + 2.47·57-s − 11.2·59-s − 3.52·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.22·7-s + 0.333·9-s + 0.342·13-s − 0.258·15-s − 1.08·17-s + 0.567·19-s − 0.706·21-s + 0.834·23-s + 0.200·25-s + 0.192·27-s − 0.513·29-s + 0.179·31-s + 0.546·35-s + 1.51·37-s + 0.197·39-s − 0.937·41-s + 0.376·43-s − 0.149·45-s + 1.30·47-s + 0.496·49-s − 0.626·51-s − 0.0648·53-s + 0.327·57-s − 1.46·59-s − 0.451·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 - 4.76T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 2.76T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−7.51708990473669529663731105131, −6.89117683122809147729774235816, −6.32503515894003481708323944250, −5.49720938395164302329753284660, −4.48372358576698542009860440303, −3.87872969493734445842300610872, −3.07418611720367838147622209905, −2.53858544188524856856524432084, −1.24125579237863329761189077616, 0,
1.24125579237863329761189077616, 2.53858544188524856856524432084, 3.07418611720367838147622209905, 3.87872969493734445842300610872, 4.48372358576698542009860440303, 5.49720938395164302329753284660, 6.32503515894003481708323944250, 6.89117683122809147729774235816, 7.51708990473669529663731105131
