L(s) = 1 | + 2-s − 3-s + 4-s − 1.44·5-s − 6-s + 2.70·7-s + 8-s + 9-s − 1.44·10-s − 2.78·11-s − 12-s − 2.73·13-s + 2.70·14-s + 1.44·15-s + 16-s + 17-s + 18-s − 1.43·19-s − 1.44·20-s − 2.70·21-s − 2.78·22-s − 3.73·23-s − 24-s − 2.91·25-s − 2.73·26-s − 27-s + 2.70·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.645·5-s − 0.408·6-s + 1.02·7-s + 0.353·8-s + 0.333·9-s − 0.456·10-s − 0.839·11-s − 0.288·12-s − 0.759·13-s + 0.723·14-s + 0.372·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.328·19-s − 0.322·20-s − 0.591·21-s − 0.593·22-s − 0.779·23-s − 0.204·24-s − 0.583·25-s − 0.536·26-s − 0.192·27-s + 0.511·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.163328064\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.163328064\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 + 1.44T + 5T^{2} \) |
| 7 | \( 1 - 2.70T + 7T^{2} \) |
| 11 | \( 1 + 2.78T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 19 | \( 1 + 1.43T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 - 4.22T + 29T^{2} \) |
| 31 | \( 1 + 2.26T + 31T^{2} \) |
| 37 | \( 1 - 8.13T + 37T^{2} \) |
| 41 | \( 1 - 8.45T + 41T^{2} \) |
| 43 | \( 1 - 4.40T + 43T^{2} \) |
| 47 | \( 1 - 0.367T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 61 | \( 1 - 3.13T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 - 4.00T + 73T^{2} \) |
| 79 | \( 1 - 0.156T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 5.89T + 97T^{2} \) |
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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
−7.81500823951721055333128762846, −7.49296015939801788093783748772, −6.59408878198592379838530587943, −5.67057026873773660726858339271, −5.26066235480245761218188633997, −4.35756316078527666570542545901, −4.05163203081392821602163528513, −2.74313169953530004347845190297, −2.03160663746220510902694884639, −0.70332761102853930559415672841,
0.70332761102853930559415672841, 2.03160663746220510902694884639, 2.74313169953530004347845190297, 4.05163203081392821602163528513, 4.35756316078527666570542545901, 5.26066235480245761218188633997, 5.67057026873773660726858339271, 6.59408878198592379838530587943, 7.49296015939801788093783748772, 7.81500823951721055333128762846