| L(s) = 1 | − 1.33·3-s + 18.4·5-s − 10.3·7-s − 25.2·9-s − 50.4·11-s − 61.8·13-s − 24.5·15-s + 68.1·17-s + 19·19-s + 13.8·21-s − 145.·23-s + 215.·25-s + 69.5·27-s + 42.6·29-s − 91.6·31-s + 67.1·33-s − 191.·35-s − 400.·37-s + 82.3·39-s − 123.·41-s − 449.·43-s − 465.·45-s + 453.·47-s − 235.·49-s − 90.7·51-s + 437.·53-s − 930.·55-s + ⋯ |
| L(s) = 1 | − 0.256·3-s + 1.64·5-s − 0.560·7-s − 0.934·9-s − 1.38·11-s − 1.31·13-s − 0.422·15-s + 0.971·17-s + 0.229·19-s + 0.143·21-s − 1.32·23-s + 1.72·25-s + 0.495·27-s + 0.272·29-s − 0.531·31-s + 0.354·33-s − 0.925·35-s − 1.78·37-s + 0.338·39-s − 0.469·41-s − 1.59·43-s − 1.54·45-s + 1.40·47-s − 0.685·49-s − 0.249·51-s + 1.13·53-s − 2.28·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 1.33T + 27T^{2} \) |
| 5 | \( 1 - 18.4T + 125T^{2} \) |
| 7 | \( 1 + 10.3T + 343T^{2} \) |
| 11 | \( 1 + 50.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.1T + 4.91e3T^{2} \) |
| 23 | \( 1 + 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 42.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 91.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 400.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 123.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 449.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 453.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 437.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 159.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 476.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 629.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 471.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 725.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 726.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 468.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 891.T + 9.12e5T^{2} \) |
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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
−10.26763580195074676103040040632, −10.22162387804147810646705125220, −9.132516529612352791773007412671, −7.938278985992462597987326589712, −6.70556912393122874192339818158, −5.55399477507143820531959854976, −5.25541532288174959219124090638, −3.04479141633118990360676554759, −2.08914211250791010900706832071, 0,
2.08914211250791010900706832071, 3.04479141633118990360676554759, 5.25541532288174959219124090638, 5.55399477507143820531959854976, 6.70556912393122874192339818158, 7.938278985992462597987326589712, 9.132516529612352791773007412671, 10.22162387804147810646705125220, 10.26763580195074676103040040632
