| L(s) = 1 | − 1.11·2-s − 0.936·3-s − 0.767·4-s + 1.03·6-s − 2.02·7-s + 3.07·8-s − 2.12·9-s + 0.335·11-s + 0.718·12-s − 1.60·13-s + 2.24·14-s − 1.87·16-s − 0.0940·17-s + 2.35·18-s − 8.15·19-s + 1.89·21-s − 0.372·22-s − 8.72·23-s − 2.87·24-s + 1.78·26-s + 4.79·27-s + 1.55·28-s − 2.89·29-s − 0.472·31-s − 4.06·32-s − 0.314·33-s + 0.104·34-s + ⋯ |
| L(s) = 1 | − 0.785·2-s − 0.540·3-s − 0.383·4-s + 0.424·6-s − 0.765·7-s + 1.08·8-s − 0.707·9-s + 0.101·11-s + 0.207·12-s − 0.445·13-s + 0.600·14-s − 0.468·16-s − 0.0228·17-s + 0.555·18-s − 1.86·19-s + 0.413·21-s − 0.0794·22-s − 1.81·23-s − 0.587·24-s + 0.349·26-s + 0.923·27-s + 0.293·28-s − 0.537·29-s − 0.0848·31-s − 0.718·32-s − 0.0547·33-s + 0.0179·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01910151995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01910151995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.11T + 2T^{2} \) |
| 3 | \( 1 + 0.936T + 3T^{2} \) |
| 7 | \( 1 + 2.02T + 7T^{2} \) |
| 11 | \( 1 - 0.335T + 11T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 17 | \( 1 + 0.0940T + 17T^{2} \) |
| 19 | \( 1 + 8.15T + 19T^{2} \) |
| 23 | \( 1 + 8.72T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 + 0.472T + 31T^{2} \) |
| 37 | \( 1 - 0.470T + 37T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 + 8.33T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 5.02T + 53T^{2} \) |
| 59 | \( 1 + 5.99T + 59T^{2} \) |
| 61 | \( 1 + 2.64T + 61T^{2} \) |
| 67 | \( 1 - 4.48T + 67T^{2} \) |
| 71 | \( 1 + 9.82T + 71T^{2} \) |
| 73 | \( 1 - 7.64T + 73T^{2} \) |
| 79 | \( 1 + 7.18T + 79T^{2} \) |
| 83 | \( 1 - 0.911T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
| 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
−8.315781456857517151767015117769, −7.49083467019002103614288072364, −6.59723075867240107646304676031, −6.08169001352897430277151718949, −5.29831010572650790680516209833, −4.40633601021325491807561481517, −3.78423585570270289441934310777, −2.61319665317312676299937511527, −1.65401114296853387944439961521, −0.082376628974448073436312467483,
0.082376628974448073436312467483, 1.65401114296853387944439961521, 2.61319665317312676299937511527, 3.78423585570270289441934310777, 4.40633601021325491807561481517, 5.29831010572650790680516209833, 6.08169001352897430277151718949, 6.59723075867240107646304676031, 7.49083467019002103614288072364, 8.315781456857517151767015117769
