| L(s) = 1 | − 2-s + 3.18·3-s + 4-s + 2.38·5-s − 3.18·6-s + 0.206·7-s − 8-s + 7.17·9-s − 2.38·10-s + 3.21·11-s + 3.18·12-s + 1.03·13-s − 0.206·14-s + 7.62·15-s + 16-s + 8.06·17-s − 7.17·18-s − 4.21·19-s + 2.38·20-s + 0.657·21-s − 3.21·22-s − 23-s − 3.18·24-s + 0.708·25-s − 1.03·26-s + 13.3·27-s + 0.206·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.84·3-s + 0.5·4-s + 1.06·5-s − 1.30·6-s + 0.0779·7-s − 0.353·8-s + 2.39·9-s − 0.755·10-s + 0.967·11-s + 0.920·12-s + 0.287·13-s − 0.0550·14-s + 1.96·15-s + 0.250·16-s + 1.95·17-s − 1.69·18-s − 0.966·19-s + 0.534·20-s + 0.143·21-s − 0.684·22-s − 0.208·23-s − 0.651·24-s + 0.141·25-s − 0.203·26-s + 2.56·27-s + 0.0389·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.291041516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.291041516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 - 2.38T + 5T^{2} \) |
| 7 | \( 1 - 0.206T + 7T^{2} \) |
| 11 | \( 1 - 3.21T + 11T^{2} \) |
| 13 | \( 1 - 1.03T + 13T^{2} \) |
| 17 | \( 1 - 8.06T + 17T^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 29 | \( 1 + 5.31T + 29T^{2} \) |
| 31 | \( 1 + 5.93T + 31T^{2} \) |
| 37 | \( 1 + 1.87T + 37T^{2} \) |
| 41 | \( 1 - 1.74T + 41T^{2} \) |
| 43 | \( 1 + 1.71T + 43T^{2} \) |
| 47 | \( 1 - 9.22T + 47T^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 - 4.09T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 4.62T + 79T^{2} \) |
| 83 | \( 1 - 2.38T + 83T^{2} \) |
| 89 | \( 1 + 2.72T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.178309373138938839691401841235, −7.61756605244130783251034580965, −6.93368617961433672415304408350, −6.11866249054625012544983993532, −5.34494564317109919382773658570, −3.97297552494899046126950215270, −3.54132798679914345603455756580, −2.56809181132849611286096606185, −1.82484121234885802537720183086, −1.27866131143588951428100519494,
1.27866131143588951428100519494, 1.82484121234885802537720183086, 2.56809181132849611286096606185, 3.54132798679914345603455756580, 3.97297552494899046126950215270, 5.34494564317109919382773658570, 6.11866249054625012544983993532, 6.93368617961433672415304408350, 7.61756605244130783251034580965, 8.178309373138938839691401841235
