L(s) = 1 | + (0.326 − 1.85i)3-s + (−2.37 − 0.866i)9-s + (0.173 − 0.300i)11-s + (0.939 − 0.342i)17-s + (−0.766 + 0.642i)19-s + (0.173 + 0.984i)25-s + (−1.43 + 2.49i)27-s + (−0.5 − 0.419i)33-s + (0.266 − 1.50i)41-s + (0.766 + 0.642i)43-s + (−0.5 + 0.866i)49-s + (−0.326 − 1.85i)51-s + (0.939 + 1.62i)57-s + (0.326 − 0.118i)59-s + (1.43 + 0.524i)67-s + ⋯ |
L(s) = 1 | + (0.326 − 1.85i)3-s + (−2.37 − 0.866i)9-s + (0.173 − 0.300i)11-s + (0.939 − 0.342i)17-s + (−0.766 + 0.642i)19-s + (0.173 + 0.984i)25-s + (−1.43 + 2.49i)27-s + (−0.5 − 0.419i)33-s + (0.266 − 1.50i)41-s + (0.766 + 0.642i)43-s + (−0.5 + 0.866i)49-s + (−0.326 − 1.85i)51-s + (0.939 + 1.62i)57-s + (0.326 − 0.118i)59-s + (1.43 + 0.524i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9705197867\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9705197867\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (0.766 - 0.642i)T \) |
good | 3 | \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \) |
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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
−10.84776614561509628390077135239, −9.494030069448359496065890732319, −8.585981900216871770114248886632, −7.83347785922591937469568122457, −7.14560303643530164893683453678, −6.24707139947005398655872817425, −5.44905249053804748627334394089, −3.60732893172876801000684977023, −2.45645617869853550690441822292, −1.23846374553900447444910465848,
2.57086109078144494480193404647, 3.68678506015844732674513004158, 4.50057578018201204885039798047, 5.34286322463509042510694121640, 6.44315153358105333983504348681, 7.952131748532381610905778392601, 8.693697641278356447360767733568, 9.539986994179437945453412388898, 10.16363206862486454347231373740, 10.84968925644618796413075463288