L(s) = 1 | + 3.18·3-s − 14.3·5-s − 17.1·7-s − 16.8·9-s − 3.82·11-s − 4.03·13-s − 45.5·15-s + 17·17-s + 36.1·19-s − 54.5·21-s − 161.·23-s + 80.1·25-s − 139.·27-s + 94.5·29-s − 40.4·31-s − 12.1·33-s + 245.·35-s − 9.76·37-s − 12.8·39-s − 387.·41-s + 169.·43-s + 241.·45-s − 284.·47-s − 49.1·49-s + 54.0·51-s + 602.·53-s + 54.7·55-s + ⋯ |
L(s) = 1 | + 0.612·3-s − 1.28·5-s − 0.925·7-s − 0.625·9-s − 0.104·11-s − 0.0861·13-s − 0.784·15-s + 0.242·17-s + 0.436·19-s − 0.566·21-s − 1.46·23-s + 0.641·25-s − 0.994·27-s + 0.605·29-s − 0.234·31-s − 0.0641·33-s + 1.18·35-s − 0.0434·37-s − 0.0527·39-s − 1.47·41-s + 0.602·43-s + 0.800·45-s − 0.883·47-s − 0.143·49-s + 0.148·51-s + 1.56·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9568539228\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9568539228\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
good | 3 | \( 1 - 3.18T + 27T^{2} \) |
| 5 | \( 1 + 14.3T + 125T^{2} \) |
| 7 | \( 1 + 17.1T + 343T^{2} \) |
| 11 | \( 1 + 3.82T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.03T + 2.19e3T^{2} \) |
| 19 | \( 1 - 36.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 94.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 9.76T + 5.06e4T^{2} \) |
| 41 | \( 1 + 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 169.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 284.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 602.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 16.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 784.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 270.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 936.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 470.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.37e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 601.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 117.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 279.T + 9.12e5T^{2} \) |
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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
−9.477194521524479446299861330330, −8.433005572929364427912428806783, −8.035076601314980965001782932616, −7.13560152957443004413013992496, −6.21358542668734878802660963294, −5.13224700534912523840359676066, −3.81817686736956717698990815839, −3.41112062737171739424299952140, −2.30813872066206093407203356991, −0.46963933834069887154160048499,
0.46963933834069887154160048499, 2.30813872066206093407203356991, 3.41112062737171739424299952140, 3.81817686736956717698990815839, 5.13224700534912523840359676066, 6.21358542668734878802660963294, 7.13560152957443004413013992496, 8.035076601314980965001782932616, 8.433005572929364427912428806783, 9.477194521524479446299861330330