| L(s) = 1 | − 0.689·3-s − 0.991·5-s + 0.742·7-s − 2.52·9-s − 3.99·11-s − 2.89·13-s + 0.682·15-s − 7.23·17-s − 19-s − 0.511·21-s + 3.71·23-s − 4.01·25-s + 3.80·27-s − 7.98·29-s + 6.55·31-s + 2.75·33-s − 0.735·35-s − 6.40·37-s + 1.99·39-s + 8.21·41-s − 8.52·43-s + 2.50·45-s + 2.16·47-s − 6.44·49-s + 4.98·51-s − 12.8·53-s + 3.95·55-s + ⋯ |
| L(s) = 1 | − 0.397·3-s − 0.443·5-s + 0.280·7-s − 0.841·9-s − 1.20·11-s − 0.804·13-s + 0.176·15-s − 1.75·17-s − 0.229·19-s − 0.111·21-s + 0.774·23-s − 0.803·25-s + 0.732·27-s − 1.48·29-s + 1.17·31-s + 0.479·33-s − 0.124·35-s − 1.05·37-s + 0.319·39-s + 1.28·41-s − 1.29·43-s + 0.373·45-s + 0.315·47-s − 0.921·49-s + 0.698·51-s − 1.76·53-s + 0.533·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4208871603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4208871603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 0.689T + 3T^{2} \) |
| 5 | \( 1 + 0.991T + 5T^{2} \) |
| 7 | \( 1 - 0.742T + 7T^{2} \) |
| 11 | \( 1 + 3.99T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 23 | \( 1 - 3.71T + 23T^{2} \) |
| 29 | \( 1 + 7.98T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 + 6.40T + 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 + 8.52T + 43T^{2} \) |
| 47 | \( 1 - 2.16T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 0.586T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 4.29T + 73T^{2} \) |
| 83 | \( 1 + 4.30T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 4.46T + 97T^{2} \) |
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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
−8.104046055736525879924131343574, −7.40150219286159001584454670144, −6.68292936948514256704771215502, −5.91082504565710871954682194086, −5.04640193135806031645084083575, −4.72694654073920001807729959265, −3.64740158774187000193485493949, −2.67520624234587528381861443431, −2.02941485807305952435593506312, −0.32260182788827847570424674483,
0.32260182788827847570424674483, 2.02941485807305952435593506312, 2.67520624234587528381861443431, 3.64740158774187000193485493949, 4.72694654073920001807729959265, 5.04640193135806031645084083575, 5.91082504565710871954682194086, 6.68292936948514256704771215502, 7.40150219286159001584454670144, 8.104046055736525879924131343574
