| L(s) = 1 | + 3·3-s − 5·5-s − 26.3·7-s + 9·9-s + 20·11-s − 40.3·13-s − 15·15-s + 101.·17-s + 29.6·19-s − 79.0·21-s + 22.3·23-s + 25·25-s + 27·27-s + 187.·29-s + 293.·31-s + 60·33-s + 131.·35-s − 1.77·37-s − 121.·39-s + 229.·41-s + 146.·43-s − 45·45-s − 538.·47-s + 351.·49-s + 303.·51-s + 432.·53-s − 100·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.42·7-s + 0.333·9-s + 0.548·11-s − 0.860·13-s − 0.258·15-s + 1.44·17-s + 0.357·19-s − 0.821·21-s + 0.202·23-s + 0.200·25-s + 0.192·27-s + 1.20·29-s + 1.69·31-s + 0.316·33-s + 0.636·35-s − 0.00788·37-s − 0.497·39-s + 0.873·41-s + 0.519·43-s − 0.149·45-s − 1.67·47-s + 1.02·49-s + 0.832·51-s + 1.11·53-s − 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.878397836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.878397836\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 26.3T + 343T^{2} \) |
| 11 | \( 1 - 20T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 29.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 22.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 293.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 1.77T + 5.06e4T^{2} \) |
| 41 | \( 1 - 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 146.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 538.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 432.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 502.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 937.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 483.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 397.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 56.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 327.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 473.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 436.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.42e3T + 9.12e5T^{2} \) |
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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
−10.13254441819733875365628237018, −9.848017801791531731264736708663, −8.843653158631719744454767642130, −7.82974908836104763408757624992, −6.97944477624608067420526082003, −6.07212395300377907053598849885, −4.66855025925024018782452245963, −3.47336047385439846872753620141, −2.75729727742228247318676678477, −0.841081131810377554802056908417,
0.841081131810377554802056908417, 2.75729727742228247318676678477, 3.47336047385439846872753620141, 4.66855025925024018782452245963, 6.07212395300377907053598849885, 6.97944477624608067420526082003, 7.82974908836104763408757624992, 8.843653158631719744454767642130, 9.848017801791531731264736708663, 10.13254441819733875365628237018
