L(s) = 1 | + 2-s − 0.111·3-s + 4-s − 1.58·5-s − 0.111·6-s + 7-s + 8-s − 2.98·9-s − 1.58·10-s − 2.72·11-s − 0.111·12-s − 0.765·13-s + 14-s + 0.177·15-s + 16-s + 2.70·17-s − 2.98·18-s + 4.75·19-s − 1.58·20-s − 0.111·21-s − 2.72·22-s + 2.13·23-s − 0.111·24-s − 2.47·25-s − 0.765·26-s + 0.668·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.0644·3-s + 0.5·4-s − 0.711·5-s − 0.0455·6-s + 0.377·7-s + 0.353·8-s − 0.995·9-s − 0.502·10-s − 0.821·11-s − 0.0322·12-s − 0.212·13-s + 0.267·14-s + 0.0458·15-s + 0.250·16-s + 0.657·17-s − 0.704·18-s + 1.09·19-s − 0.355·20-s − 0.0243·21-s − 0.580·22-s + 0.444·23-s − 0.0227·24-s − 0.494·25-s − 0.150·26-s + 0.128·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.307095078\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.307095078\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 + 0.111T + 3T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 + 0.765T + 13T^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 19 | \( 1 - 4.75T + 19T^{2} \) |
| 23 | \( 1 - 2.13T + 23T^{2} \) |
| 29 | \( 1 + 0.231T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 + 5.36T + 41T^{2} \) |
| 43 | \( 1 - 4.84T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 0.449T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 0.241T + 73T^{2} \) |
| 79 | \( 1 - 9.92T + 79T^{2} \) |
| 83 | \( 1 - 6.57T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.977050512288332929374304495226, −7.49008572584998577859491483517, −6.55461984141713162116157706274, −5.80788762145560800037707359721, −5.04966903604313856882411807276, −4.68771967514536747478885630396, −3.41935946636239329333573546153, −3.11143759485759125824821020653, −2.06716528575845230778575011519, −0.70257501696985449642878469409,
0.70257501696985449642878469409, 2.06716528575845230778575011519, 3.11143759485759125824821020653, 3.41935946636239329333573546153, 4.68771967514536747478885630396, 5.04966903604313856882411807276, 5.80788762145560800037707359721, 6.55461984141713162116157706274, 7.49008572584998577859491483517, 7.977050512288332929374304495226