| L(s) = 1 | − 2-s − 1.53·3-s + 4-s + 2·5-s + 1.53·6-s + 2.69·7-s − 8-s − 0.652·9-s − 2·10-s + 3.18·11-s − 1.53·12-s + 5.75·13-s − 2.69·14-s − 3.06·15-s + 16-s − 6.51·17-s + 0.652·18-s + 2·20-s − 4.12·21-s − 3.18·22-s − 0.694·23-s + 1.53·24-s − 25-s − 5.75·26-s + 5.59·27-s + 2.69·28-s + 2.82·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.884·3-s + 0.5·4-s + 0.894·5-s + 0.625·6-s + 1.01·7-s − 0.353·8-s − 0.217·9-s − 0.632·10-s + 0.960·11-s − 0.442·12-s + 1.59·13-s − 0.720·14-s − 0.791·15-s + 0.250·16-s − 1.58·17-s + 0.153·18-s + 0.447·20-s − 0.900·21-s − 0.679·22-s − 0.144·23-s + 0.312·24-s − 0.200·25-s − 1.12·26-s + 1.07·27-s + 0.509·28-s + 0.524·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.106992466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106992466\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.53T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 - 5.75T + 13T^{2} \) |
| 17 | \( 1 + 6.51T + 17T^{2} \) |
| 23 | \( 1 + 0.694T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 2.45T + 31T^{2} \) |
| 37 | \( 1 - 4.36T + 37T^{2} \) |
| 41 | \( 1 - 0.347T + 41T^{2} \) |
| 43 | \( 1 - 6.06T + 43T^{2} \) |
| 47 | \( 1 - 7.88T + 47T^{2} \) |
| 53 | \( 1 + 8.21T + 53T^{2} \) |
| 59 | \( 1 + 0.573T + 59T^{2} \) |
| 61 | \( 1 - 2.93T + 61T^{2} \) |
| 67 | \( 1 - 4.95T + 67T^{2} \) |
| 71 | \( 1 - 8.45T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 - 9.06T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 + 7.73T + 89T^{2} \) |
| 97 | \( 1 - 0.347T + 97T^{2} \) |
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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
−10.70608456391033715892821252605, −9.418073384910046762245221473982, −8.816405147039691545437046672649, −8.010489372792661589567420402807, −6.55166341077350550769152703559, −6.22164820933960032027253201021, −5.22886524434935809283915471190, −4.02210022282092120725833698475, −2.20934183984285395708508559718, −1.10910445578557446570450185485,
1.10910445578557446570450185485, 2.20934183984285395708508559718, 4.02210022282092120725833698475, 5.22886524434935809283915471190, 6.22164820933960032027253201021, 6.55166341077350550769152703559, 8.010489372792661589567420402807, 8.816405147039691545437046672649, 9.418073384910046762245221473982, 10.70608456391033715892821252605
