| L(s) = 1 | + 2-s − 2.03·3-s + 4-s − 5-s − 2.03·6-s + 1.56·7-s + 8-s + 1.13·9-s − 10-s + 5.44·11-s − 2.03·12-s + 3.46·13-s + 1.56·14-s + 2.03·15-s + 16-s − 7.61·17-s + 1.13·18-s + 4.41·19-s − 20-s − 3.18·21-s + 5.44·22-s − 2.03·24-s + 25-s + 3.46·26-s + 3.78·27-s + 1.56·28-s + 0.733·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.17·3-s + 0.5·4-s − 0.447·5-s − 0.830·6-s + 0.592·7-s + 0.353·8-s + 0.379·9-s − 0.316·10-s + 1.64·11-s − 0.587·12-s + 0.961·13-s + 0.418·14-s + 0.525·15-s + 0.250·16-s − 1.84·17-s + 0.268·18-s + 1.01·19-s − 0.223·20-s − 0.695·21-s + 1.16·22-s − 0.415·24-s + 0.200·25-s + 0.679·26-s + 0.729·27-s + 0.296·28-s + 0.136·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.264336242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.264336242\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.03T + 3T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 5.44T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 7.61T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 29 | \( 1 - 0.733T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 2.70T + 41T^{2} \) |
| 43 | \( 1 + 2.10T + 43T^{2} \) |
| 47 | \( 1 - 7.35T + 47T^{2} \) |
| 53 | \( 1 - 5.89T + 53T^{2} \) |
| 59 | \( 1 - 2.61T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 3.38T + 67T^{2} \) |
| 71 | \( 1 + 1.90T + 71T^{2} \) |
| 73 | \( 1 - 6.83T + 73T^{2} \) |
| 79 | \( 1 + 1.58T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.124788387979958476631525195736, −7.11890643855307982335522706648, −6.57220225678931291512179444702, −6.04730814516764182332379786014, −5.30863933996833314229548427724, −4.39098380161745524404500819251, −4.12131271513977731770547915083, −3.03666177589569425066414900498, −1.72847678839236236780399777877, −0.820462928874519947577982220929,
0.820462928874519947577982220929, 1.72847678839236236780399777877, 3.03666177589569425066414900498, 4.12131271513977731770547915083, 4.39098380161745524404500819251, 5.30863933996833314229548427724, 6.04730814516764182332379786014, 6.57220225678931291512179444702, 7.11890643855307982335522706648, 8.124788387979958476631525195736
