| L(s) = 1 | + 18.2·3-s − 97.7·5-s − 139.·7-s + 91.2·9-s − 533.·11-s − 675.·13-s − 1.78e3·15-s − 268.·17-s + 2.64e3·19-s − 2.55e3·21-s − 794.·23-s + 6.42e3·25-s − 2.77e3·27-s − 841·29-s + 4.23e3·31-s − 9.74e3·33-s + 1.36e4·35-s − 2.68e3·37-s − 1.23e4·39-s + 1.39e3·41-s + 2.38e4·43-s − 8.91e3·45-s − 1.12e4·47-s + 2.66e3·49-s − 4.91e3·51-s − 3.39e3·53-s + 5.20e4·55-s + ⋯ |
| L(s) = 1 | + 1.17·3-s − 1.74·5-s − 1.07·7-s + 0.375·9-s − 1.32·11-s − 1.10·13-s − 2.05·15-s − 0.225·17-s + 1.68·19-s − 1.26·21-s − 0.313·23-s + 2.05·25-s − 0.732·27-s − 0.185·29-s + 0.790·31-s − 1.55·33-s + 1.88·35-s − 0.322·37-s − 1.30·39-s + 0.129·41-s + 1.96·43-s − 0.656·45-s − 0.744·47-s + 0.158·49-s − 0.264·51-s − 0.166·53-s + 2.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9371138542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9371138542\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 841T \) |
| good | 3 | \( 1 - 18.2T + 243T^{2} \) |
| 5 | \( 1 + 97.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 139.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 533.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 675.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 268.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.64e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 794.T + 6.43e6T^{2} \) |
| 31 | \( 1 - 4.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.12e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.39e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.78e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.57e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.99e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.99e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.38e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.30e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.38e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.76e5T + 8.58e9T^{2} \) |
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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
−10.02408863621268333985551433746, −9.308168887186927059807281642469, −8.197851445685555052860955277628, −7.69178087351033369236842615536, −7.02439188296806741820835992914, −5.34367901821058604854183517735, −4.13782562735385345803117441009, −3.15354516712851817614347690593, −2.65868286471109343450145384662, −0.43765192435341719342969391476,
0.43765192435341719342969391476, 2.65868286471109343450145384662, 3.15354516712851817614347690593, 4.13782562735385345803117441009, 5.34367901821058604854183517735, 7.02439188296806741820835992914, 7.69178087351033369236842615536, 8.197851445685555052860955277628, 9.308168887186927059807281642469, 10.02408863621268333985551433746
