L(s) = 1 | + 2-s + 2.14·3-s + 4-s + 2.45·5-s + 2.14·6-s + 7-s + 8-s + 1.60·9-s + 2.45·10-s + 4.66·11-s + 2.14·12-s − 1.96·13-s + 14-s + 5.26·15-s + 16-s − 1.67·17-s + 1.60·18-s + 2.45·20-s + 2.14·21-s + 4.66·22-s + 0.371·23-s + 2.14·24-s + 1.01·25-s − 1.96·26-s − 3.00·27-s + 28-s + 6.25·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.23·3-s + 0.5·4-s + 1.09·5-s + 0.875·6-s + 0.377·7-s + 0.353·8-s + 0.533·9-s + 0.775·10-s + 1.40·11-s + 0.619·12-s − 0.546·13-s + 0.267·14-s + 1.35·15-s + 0.250·16-s − 0.407·17-s + 0.377·18-s + 0.548·20-s + 0.468·21-s + 0.993·22-s + 0.0773·23-s + 0.437·24-s + 0.203·25-s − 0.386·26-s − 0.577·27-s + 0.188·28-s + 1.16·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.669723105\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.669723105\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 2.14T + 3T^{2} \) |
| 5 | \( 1 - 2.45T + 5T^{2} \) |
| 11 | \( 1 - 4.66T + 11T^{2} \) |
| 13 | \( 1 + 1.96T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 23 | \( 1 - 0.371T + 23T^{2} \) |
| 29 | \( 1 - 6.25T + 29T^{2} \) |
| 31 | \( 1 - 7.32T + 31T^{2} \) |
| 37 | \( 1 + 0.646T + 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 + 9.92T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 5.55T + 53T^{2} \) |
| 59 | \( 1 - 2.02T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 6.75T + 67T^{2} \) |
| 71 | \( 1 - 8.28T + 71T^{2} \) |
| 73 | \( 1 + 2.51T + 73T^{2} \) |
| 79 | \( 1 - 9.58T + 79T^{2} \) |
| 83 | \( 1 - 9.73T + 83T^{2} \) |
| 89 | \( 1 + 1.96T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.365525664342673369533774808413, −7.51551432131831385337815137825, −6.53670403880818796373028977922, −6.29358063496635048099684229236, −5.13909242077592013011609575318, −4.52802036620333974878902705612, −3.61512790012955804200755098787, −2.85488079451962122979833535080, −2.08730309836234602537064812882, −1.39589958429277365524211938936,
1.39589958429277365524211938936, 2.08730309836234602537064812882, 2.85488079451962122979833535080, 3.61512790012955804200755098787, 4.52802036620333974878902705612, 5.13909242077592013011609575318, 6.29358063496635048099684229236, 6.53670403880818796373028977922, 7.51551432131831385337815137825, 8.365525664342673369533774808413