| L(s) = 1 | + 2.37·2-s + 3.61·4-s + 3.83·5-s − 4.47·7-s + 3.83·8-s + 9.09·10-s − 2.92·11-s − 13-s − 10.6·14-s + 1.85·16-s − 4.74·17-s + 0.763·19-s + 13.8·20-s − 6.94·22-s + 2.92·23-s + 9.70·25-s − 2.37·26-s − 16.1·28-s + 2.92·29-s + 7.23·31-s − 3.27·32-s − 11.2·34-s − 17.1·35-s + 3.23·37-s + 1.81·38-s + 14.7·40-s − 4.74·41-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 1.80·4-s + 1.71·5-s − 1.69·7-s + 1.35·8-s + 2.87·10-s − 0.883·11-s − 0.277·13-s − 2.83·14-s + 0.463·16-s − 1.14·17-s + 0.175·19-s + 3.10·20-s − 1.48·22-s + 0.610·23-s + 1.94·25-s − 0.464·26-s − 3.05·28-s + 0.544·29-s + 1.29·31-s − 0.579·32-s − 1.92·34-s − 2.89·35-s + 0.532·37-s + 0.293·38-s + 2.32·40-s − 0.740·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.380552154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.380552154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 - 3.83T + 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 - 2.92T + 23T^{2} \) |
| 29 | \( 1 - 2.92T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 + 4.74T + 41T^{2} \) |
| 43 | \( 1 - 0.236T + 43T^{2} \) |
| 47 | \( 1 + 3.83T + 47T^{2} \) |
| 53 | \( 1 - 5.85T + 53T^{2} \) |
| 59 | \( 1 + 2.02T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 2.76T + 67T^{2} \) |
| 71 | \( 1 + 8.57T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 0.905T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−11.85759936279832507629127837917, −10.52391339053850605668585089177, −9.911852583566992499368420276486, −8.934796167837110223133659516462, −6.93799082234239821201801540092, −6.35366473119188168389952926249, −5.62168097322788202537509944600, −4.64473121299973458699659999532, −3.05783971289091392186173856254, −2.40741513953788504731764136409,
2.40741513953788504731764136409, 3.05783971289091392186173856254, 4.64473121299973458699659999532, 5.62168097322788202537509944600, 6.35366473119188168389952926249, 6.93799082234239821201801540092, 8.934796167837110223133659516462, 9.911852583566992499368420276486, 10.52391339053850605668585089177, 11.85759936279832507629127837917
