| L(s) = 1 | − 0.115·2-s + 3.28·3-s − 1.98·4-s − 0.379·6-s − 3.19·7-s + 0.461·8-s + 7.77·9-s + 1.38·11-s − 6.52·12-s + 5.87·13-s + 0.369·14-s + 3.91·16-s − 5.28·17-s − 0.899·18-s + 4.99·19-s − 10.4·21-s − 0.160·22-s − 3.07·23-s + 1.51·24-s − 0.679·26-s + 15.6·27-s + 6.35·28-s + 3.28·29-s + 0.672·31-s − 1.37·32-s + 4.56·33-s + 0.611·34-s + ⋯ |
| L(s) = 1 | − 0.0817·2-s + 1.89·3-s − 0.993·4-s − 0.155·6-s − 1.20·7-s + 0.163·8-s + 2.59·9-s + 0.419·11-s − 1.88·12-s + 1.62·13-s + 0.0988·14-s + 0.979·16-s − 1.28·17-s − 0.212·18-s + 1.14·19-s − 2.28·21-s − 0.0342·22-s − 0.640·23-s + 0.309·24-s − 0.133·26-s + 3.01·27-s + 1.20·28-s + 0.610·29-s + 0.120·31-s − 0.243·32-s + 0.794·33-s + 0.104·34-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
| \(L(1)\) |
\(\approx\) |
\(3.122178924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.122178924\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.115T + 2T^{2} \) |
| 3 | \( 1 - 3.28T + 3T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 + 5.28T + 17T^{2} \) |
| 19 | \( 1 - 4.99T + 19T^{2} \) |
| 23 | \( 1 + 3.07T + 23T^{2} \) |
| 29 | \( 1 - 3.28T + 29T^{2} \) |
| 31 | \( 1 - 0.672T + 31T^{2} \) |
| 37 | \( 1 - 3.79T + 37T^{2} \) |
| 41 | \( 1 - 0.970T + 41T^{2} \) |
| 43 | \( 1 + 7.93T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 8.45T + 53T^{2} \) |
| 59 | \( 1 - 5.70T + 59T^{2} \) |
| 61 | \( 1 - 0.717T + 61T^{2} \) |
| 67 | \( 1 + 8.81T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 8.75T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 1.18T + 97T^{2} \) |
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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
−8.306786298371722780284242594766, −7.68199274647943811927062716337, −6.67556718241314105543048148071, −6.23241649796549713590895903585, −4.90958320987750029423872147565, −4.07521071723478479893693598837, −3.53235551557354699969678916681, −3.10178665422785639102991551509, −1.94761472284328676870295021048, −0.898275218291623688770780709962,
0.898275218291623688770780709962, 1.94761472284328676870295021048, 3.10178665422785639102991551509, 3.53235551557354699969678916681, 4.07521071723478479893693598837, 4.90958320987750029423872147565, 6.23241649796549713590895903585, 6.67556718241314105543048148071, 7.68199274647943811927062716337, 8.306786298371722780284242594766
