| L(s) = 1 | + 12.3·5-s + 30.6·7-s + 0.944·11-s + 90.8·13-s + 53.4·17-s − 70.4·19-s + 97.1·23-s + 28.6·25-s − 156.·29-s − 43.0·31-s + 379.·35-s − 64.1·37-s − 85.0·41-s + 167.·43-s − 558.·47-s + 595.·49-s − 503.·53-s + 11.7·55-s − 61.7·59-s + 269.·61-s + 1.12e3·65-s − 126.·67-s − 854.·71-s + 404.·73-s + 28.9·77-s + 646.·79-s − 822.·83-s + ⋯ |
| L(s) = 1 | + 1.10·5-s + 1.65·7-s + 0.0258·11-s + 1.93·13-s + 0.761·17-s − 0.851·19-s + 0.881·23-s + 0.229·25-s − 1.00·29-s − 0.249·31-s + 1.83·35-s − 0.285·37-s − 0.324·41-s + 0.593·43-s − 1.73·47-s + 1.73·49-s − 1.30·53-s + 0.0286·55-s − 0.136·59-s + 0.564·61-s + 2.14·65-s − 0.230·67-s − 1.42·71-s + 0.648·73-s + 0.0428·77-s + 0.920·79-s − 1.08·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.402366809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.402366809\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 12.3T + 125T^{2} \) |
| 7 | \( 1 - 30.6T + 343T^{2} \) |
| 11 | \( 1 - 0.944T + 1.33e3T^{2} \) |
| 13 | \( 1 - 90.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 97.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 43.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 64.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 85.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 167.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 558.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 503.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 61.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 269.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 126.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 854.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 404.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 646.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 822.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 130.T + 9.12e5T^{2} \) |
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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
−10.29572310222271128553351973953, −9.154006084680550478453498329488, −8.488799927570949408572980668123, −7.67562700223318600500612918788, −6.37520097275615986930045233874, −5.63929263771576616145648132057, −4.76253361196303770827556639790, −3.52474836390175716225769580572, −1.92417622211677267329442039962, −1.27551607915097804608283099619,
1.27551607915097804608283099619, 1.92417622211677267329442039962, 3.52474836390175716225769580572, 4.76253361196303770827556639790, 5.63929263771576616145648132057, 6.37520097275615986930045233874, 7.67562700223318600500612918788, 8.488799927570949408572980668123, 9.154006084680550478453498329488, 10.29572310222271128553351973953
