| L(s) = 1 | − 2.03·2-s + 2.15·4-s + 3.24·5-s + 1.92·7-s − 0.314·8-s − 6.61·10-s + 11-s + 2.85·13-s − 3.92·14-s − 3.66·16-s + 2.33·17-s + 19-s + 6.99·20-s − 2.03·22-s + 2.74·23-s + 5.54·25-s − 5.81·26-s + 4.14·28-s + 0.972·29-s − 0.00551·31-s + 8.10·32-s − 4.74·34-s + 6.24·35-s + 9.67·37-s − 2.03·38-s − 1.02·40-s − 6.65·41-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.07·4-s + 1.45·5-s + 0.726·7-s − 0.111·8-s − 2.09·10-s + 0.301·11-s + 0.790·13-s − 1.04·14-s − 0.916·16-s + 0.565·17-s + 0.229·19-s + 1.56·20-s − 0.434·22-s + 0.572·23-s + 1.10·25-s − 1.13·26-s + 0.783·28-s + 0.180·29-s − 0.000989·31-s + 1.43·32-s − 0.814·34-s + 1.05·35-s + 1.58·37-s − 0.330·38-s − 0.161·40-s − 1.03·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.369607193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.369607193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.03T + 2T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 7 | \( 1 - 1.92T + 7T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 2.33T + 17T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 - 0.972T + 29T^{2} \) |
| 31 | \( 1 + 0.00551T + 31T^{2} \) |
| 37 | \( 1 - 9.67T + 37T^{2} \) |
| 41 | \( 1 + 6.65T + 41T^{2} \) |
| 43 | \( 1 - 7.99T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 + 3.97T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 1.87T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 7.57T + 97T^{2} \) |
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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
−9.319879504533633909801461997239, −8.561818173151566950789849948836, −7.937487750241571940315116968190, −7.01065264906649485739394231385, −6.19710772111175569554518685828, −5.41496415779234044336748086951, −4.38721326057825618521911314110, −2.85066113680062216193465736387, −1.73207091703478115952497059605, −1.11810863354556702352829007086,
1.11810863354556702352829007086, 1.73207091703478115952497059605, 2.85066113680062216193465736387, 4.38721326057825618521911314110, 5.41496415779234044336748086951, 6.19710772111175569554518685828, 7.01065264906649485739394231385, 7.937487750241571940315116968190, 8.561818173151566950789849948836, 9.319879504533633909801461997239
