| L(s) = 1 | − 7.91·2-s + 30.6·4-s − 79.5·5-s − 172.·7-s + 11.0·8-s + 629.·10-s − 452.·11-s − 22.7·13-s + 1.36e3·14-s − 1.06e3·16-s + 521.·17-s + 1.55e3·19-s − 2.43e3·20-s + 3.57e3·22-s + 3.46e3·23-s + 3.20e3·25-s + 179.·26-s − 5.27e3·28-s − 4.32e3·29-s − 3.98e3·31-s + 8.08e3·32-s − 4.12e3·34-s + 1.37e4·35-s + 1.00e4·37-s − 1.23e4·38-s − 879.·40-s + 1.64e4·41-s + ⋯ |
| L(s) = 1 | − 1.39·2-s + 0.956·4-s − 1.42·5-s − 1.32·7-s + 0.0610·8-s + 1.99·10-s − 1.12·11-s − 0.0373·13-s + 1.85·14-s − 1.04·16-s + 0.437·17-s + 0.990·19-s − 1.36·20-s + 1.57·22-s + 1.36·23-s + 1.02·25-s + 0.0522·26-s − 1.27·28-s − 0.954·29-s − 0.745·31-s + 1.39·32-s − 0.612·34-s + 1.89·35-s + 1.21·37-s − 1.38·38-s − 0.0869·40-s + 1.52·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 + 7.91T + 32T^{2} \) |
| 5 | \( 1 + 79.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 172.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 452.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 22.7T + 3.71e5T^{2} \) |
| 17 | \( 1 - 521.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.55e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.46e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.00e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.64e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 2.41e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.89e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.30e4T + 8.58e9T^{2} \) |
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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
−9.779903950267240900377809969477, −9.279937719801944847917218491023, −8.065073071496277017586365033574, −7.58782089790949496625056632317, −6.76124529411905240247977941765, −5.20943976294467414155042637793, −3.75613920266145785460669643623, −2.75107391262583876002159612192, −0.825377433133850514856217092949, 0,
0.825377433133850514856217092949, 2.75107391262583876002159612192, 3.75613920266145785460669643623, 5.20943976294467414155042637793, 6.76124529411905240247977941765, 7.58782089790949496625056632317, 8.065073071496277017586365033574, 9.279937719801944847917218491023, 9.779903950267240900377809969477
