| L(s) = 1 | − 3-s + 2.48·5-s − 2.46·7-s + 9-s + 4.30·11-s + 0.0484·13-s − 2.48·15-s − 4.02·17-s + 3.53·19-s + 2.46·21-s + 0.861·23-s + 1.18·25-s − 27-s − 6.05·29-s − 5.71·31-s − 4.30·33-s − 6.11·35-s − 10.3·37-s − 0.0484·39-s − 10.0·41-s − 6.87·43-s + 2.48·45-s + 12.5·47-s − 0.944·49-s + 4.02·51-s + 9.81·53-s + 10.7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.11·5-s − 0.930·7-s + 0.333·9-s + 1.29·11-s + 0.0134·13-s − 0.641·15-s − 0.975·17-s + 0.810·19-s + 0.536·21-s + 0.179·23-s + 0.236·25-s − 0.192·27-s − 1.12·29-s − 1.02·31-s − 0.749·33-s − 1.03·35-s − 1.70·37-s − 0.00776·39-s − 1.56·41-s − 1.04·43-s + 0.370·45-s + 1.83·47-s − 0.134·49-s + 0.563·51-s + 1.34·53-s + 1.44·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 - 2.48T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 - 0.0484T + 13T^{2} \) |
| 17 | \( 1 + 4.02T + 17T^{2} \) |
| 19 | \( 1 - 3.53T + 19T^{2} \) |
| 23 | \( 1 - 0.861T + 23T^{2} \) |
| 29 | \( 1 + 6.05T + 29T^{2} \) |
| 31 | \( 1 + 5.71T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 6.87T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 9.81T + 53T^{2} \) |
| 59 | \( 1 + 6.83T + 59T^{2} \) |
| 61 | \( 1 + 8.26T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 - 0.747T + 71T^{2} \) |
| 73 | \( 1 - 5.86T + 73T^{2} \) |
| 79 | \( 1 - 1.78T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.43181908766816543695533977977, −6.76732014308557993232979209529, −6.39684933284950522111387064842, −5.61359756249327307741214537252, −5.08966228011436237863211334572, −3.93907165181208789443217240433, −3.35864790962834437487493119558, −2.11407011347042820852912202731, −1.41202114077181460183705442427, 0,
1.41202114077181460183705442427, 2.11407011347042820852912202731, 3.35864790962834437487493119558, 3.93907165181208789443217240433, 5.08966228011436237863211334572, 5.61359756249327307741214537252, 6.39684933284950522111387064842, 6.76732014308557993232979209529, 7.43181908766816543695533977977
