L(s) = 1 | + 2.27·2-s + 3-s + 3.17·4-s + 1.23·5-s + 2.27·6-s + 2.67·8-s + 9-s + 2.81·10-s + 6.12·11-s + 3.17·12-s − 0.910·13-s + 1.23·15-s − 0.266·16-s − 4.90·17-s + 2.27·18-s + 6.90·19-s + 3.93·20-s + 13.9·22-s + 23-s + 2.67·24-s − 3.46·25-s − 2.07·26-s + 27-s + 3.36·29-s + 2.81·30-s − 5.43·31-s − 5.95·32-s + ⋯ |
L(s) = 1 | + 1.60·2-s + 0.577·3-s + 1.58·4-s + 0.553·5-s + 0.928·6-s + 0.945·8-s + 0.333·9-s + 0.891·10-s + 1.84·11-s + 0.916·12-s − 0.252·13-s + 0.319·15-s − 0.0665·16-s − 1.19·17-s + 0.536·18-s + 1.58·19-s + 0.879·20-s + 2.97·22-s + 0.208·23-s + 0.545·24-s − 0.693·25-s − 0.406·26-s + 0.192·27-s + 0.623·29-s + 0.514·30-s − 0.976·31-s − 1.05·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.077873030\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.077873030\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 - 6.12T + 11T^{2} \) |
| 13 | \( 1 + 0.910T + 13T^{2} \) |
| 17 | \( 1 + 4.90T + 17T^{2} \) |
| 19 | \( 1 - 6.90T + 19T^{2} \) |
| 29 | \( 1 - 3.36T + 29T^{2} \) |
| 31 | \( 1 + 5.43T + 31T^{2} \) |
| 37 | \( 1 - 6.54T + 37T^{2} \) |
| 41 | \( 1 - 2.52T + 41T^{2} \) |
| 43 | \( 1 + 0.231T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 - 0.754T + 59T^{2} \) |
| 61 | \( 1 + 3.57T + 61T^{2} \) |
| 67 | \( 1 + 4.49T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 4.27T + 73T^{2} \) |
| 79 | \( 1 + 0.222T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 0.559T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−8.726960391689294818736985341956, −7.57765789395662030327769769063, −6.78293903168099602816862702777, −6.32412943229409900587810880680, −5.48393987890876123543205757930, −4.64346596475739038096641790125, −3.95662818310493623214164778110, −3.27285576435235315994033966954, −2.34341960124748917091175001033, −1.42918714266862521344608951973,
1.42918714266862521344608951973, 2.34341960124748917091175001033, 3.27285576435235315994033966954, 3.95662818310493623214164778110, 4.64346596475739038096641790125, 5.48393987890876123543205757930, 6.32412943229409900587810880680, 6.78293903168099602816862702777, 7.57765789395662030327769769063, 8.726960391689294818736985341956