| L(s) = 1 | − 2.53·2-s + 3-s + 4.42·4-s + 5-s − 2.53·6-s + 3.85·7-s − 6.15·8-s + 9-s − 2.53·10-s − 4.87·11-s + 4.42·12-s + 0.512·13-s − 9.78·14-s + 15-s + 6.74·16-s + 5.81·17-s − 2.53·18-s − 0.722·19-s + 4.42·20-s + 3.85·21-s + 12.3·22-s + 9.55·23-s − 6.15·24-s + 25-s − 1.29·26-s + 27-s + 17.0·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.577·3-s + 2.21·4-s + 0.447·5-s − 1.03·6-s + 1.45·7-s − 2.17·8-s + 0.333·9-s − 0.801·10-s − 1.46·11-s + 1.27·12-s + 0.142·13-s − 2.61·14-s + 0.258·15-s + 1.68·16-s + 1.40·17-s − 0.597·18-s − 0.165·19-s + 0.989·20-s + 0.842·21-s + 2.63·22-s + 1.99·23-s − 1.25·24-s + 0.200·25-s − 0.254·26-s + 0.192·27-s + 3.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.520923219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520923219\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 - 0.512T + 13T^{2} \) |
| 17 | \( 1 - 5.81T + 17T^{2} \) |
| 19 | \( 1 + 0.722T + 19T^{2} \) |
| 23 | \( 1 - 9.55T + 23T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 - 9.61T + 31T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 - 5.72T + 41T^{2} \) |
| 43 | \( 1 + 0.643T + 43T^{2} \) |
| 47 | \( 1 - 1.45T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 + 0.197T + 61T^{2} \) |
| 67 | \( 1 + 4.46T + 67T^{2} \) |
| 71 | \( 1 + 7.10T + 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.243075936595328735621276825200, −7.54466077168935921797227726069, −7.31601139670305315410606740303, −6.11943475772100558739283135102, −5.32300649760253838692268605658, −4.58619256706514297137627712029, −3.04709082468854031920244076361, −2.51362335452903549213055096221, −1.57547338371211018492484248937, −0.895268207939452503954834995023,
0.895268207939452503954834995023, 1.57547338371211018492484248937, 2.51362335452903549213055096221, 3.04709082468854031920244076361, 4.58619256706514297137627712029, 5.32300649760253838692268605658, 6.11943475772100558739283135102, 7.31601139670305315410606740303, 7.54466077168935921797227726069, 8.243075936595328735621276825200
