| L(s) = 1 | − 25·5-s + 42.1·7-s + 416.·11-s + 966.·13-s + 1.83e3·17-s − 317.·19-s + 1.56e3·23-s + 625·25-s − 7.75e3·29-s − 102.·31-s − 1.05e3·35-s + 1.93e3·37-s − 7.99e3·41-s − 1.65e4·43-s + 1.86e4·47-s − 1.50e4·49-s + 1.49e4·53-s − 1.04e4·55-s + 1.98e4·59-s − 1.80e4·61-s − 2.41e4·65-s + 5.50e4·67-s + 1.12e4·71-s − 4.01e3·73-s + 1.75e4·77-s − 2.40e4·79-s + 7.05e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.325·7-s + 1.03·11-s + 1.58·13-s + 1.53·17-s − 0.201·19-s + 0.618·23-s + 0.200·25-s − 1.71·29-s − 0.0191·31-s − 0.145·35-s + 0.232·37-s − 0.742·41-s − 1.36·43-s + 1.23·47-s − 0.894·49-s + 0.732·53-s − 0.463·55-s + 0.742·59-s − 0.620·61-s − 0.709·65-s + 1.49·67-s + 0.263·71-s − 0.0881·73-s + 0.337·77-s − 0.432·79-s + 1.12·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.693540124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.693540124\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| good | 7 | \( 1 - 42.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 416.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 966.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.83e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 317.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 102.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.99e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.86e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.01e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.11e4T + 8.58e9T^{2} \) |
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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
−9.558231092951969392762027878280, −8.730755667115549615916677010986, −8.004157725220559468593713130648, −7.05221671953603613736669626367, −6.10837302477779153570364515610, −5.20405706781451050498532255341, −3.89399362627151001755098480783, −3.39856027263166861999511131821, −1.68514736068514509936462472991, −0.830344355230170535350590162482,
0.830344355230170535350590162482, 1.68514736068514509936462472991, 3.39856027263166861999511131821, 3.89399362627151001755098480783, 5.20405706781451050498532255341, 6.10837302477779153570364515610, 7.05221671953603613736669626367, 8.004157725220559468593713130648, 8.730755667115549615916677010986, 9.558231092951969392762027878280
