| L(s) = 1 | + 1.88·3-s − 1.82·5-s + 0.552·9-s − 0.945·11-s + 13-s − 3.43·15-s − 2.37·17-s + 5.75·19-s − 3.70·23-s − 1.67·25-s − 4.61·27-s + 8.67·29-s + 1.33·31-s − 1.78·33-s − 9.81·37-s + 1.88·39-s − 4.61·41-s − 3.26·43-s − 1.00·45-s − 5.26·47-s − 4.47·51-s + 9.19·53-s + 1.72·55-s + 10.8·57-s − 3.45·59-s − 8.73·61-s − 1.82·65-s + ⋯ |
| L(s) = 1 | + 1.08·3-s − 0.815·5-s + 0.184·9-s − 0.285·11-s + 0.277·13-s − 0.887·15-s − 0.576·17-s + 1.32·19-s − 0.773·23-s − 0.334·25-s − 0.887·27-s + 1.61·29-s + 0.239·31-s − 0.310·33-s − 1.61·37-s + 0.301·39-s − 0.720·41-s − 0.497·43-s − 0.150·45-s − 0.768·47-s − 0.627·51-s + 1.26·53-s + 0.232·55-s + 1.43·57-s − 0.450·59-s − 1.11·61-s − 0.226·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.88T + 3T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 11 | \( 1 + 0.945T + 11T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 19 | \( 1 - 5.75T + 19T^{2} \) |
| 23 | \( 1 + 3.70T + 23T^{2} \) |
| 29 | \( 1 - 8.67T + 29T^{2} \) |
| 31 | \( 1 - 1.33T + 31T^{2} \) |
| 37 | \( 1 + 9.81T + 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 + 5.26T + 47T^{2} \) |
| 53 | \( 1 - 9.19T + 53T^{2} \) |
| 59 | \( 1 + 3.45T + 59T^{2} \) |
| 61 | \( 1 + 8.73T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 - 9.95T + 71T^{2} \) |
| 73 | \( 1 - 6.24T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 4.40T + 83T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.076062605288078489338077182297, −7.33518611858789820270522919895, −6.63562043221881654647595961643, −5.63376947320697277583547458587, −4.79270590958529451431621706469, −3.90393846070795283299872550613, −3.29520800582624657978055457094, −2.59072585920884231930620893675, −1.50798592577087626929777752170, 0,
1.50798592577087626929777752170, 2.59072585920884231930620893675, 3.29520800582624657978055457094, 3.90393846070795283299872550613, 4.79270590958529451431621706469, 5.63376947320697277583547458587, 6.63562043221881654647595961643, 7.33518611858789820270522919895, 8.076062605288078489338077182297
