L(s) = 1 | − 1.93·2-s − 0.517·3-s + 1.73·4-s + 0.999·6-s − 0.896·7-s + 0.517·8-s − 2.73·9-s − 0.896·12-s + 4.24·13-s + 1.73·14-s − 4.46·16-s − 1.03·17-s + 5.27·18-s − 6.19·19-s + 0.464·21-s + 6.31·23-s − 0.267·24-s − 8.19·26-s + 2.96·27-s − 1.55·28-s + 6.92·29-s − 5.26·31-s + 7.58·32-s + 1.99·34-s − 4.73·36-s − 6.69·37-s + 11.9·38-s + ⋯ |
L(s) = 1 | − 1.36·2-s − 0.298·3-s + 0.866·4-s + 0.408·6-s − 0.338·7-s + 0.183·8-s − 0.910·9-s − 0.258·12-s + 1.17·13-s + 0.462·14-s − 1.11·16-s − 0.251·17-s + 1.24·18-s − 1.42·19-s + 0.101·21-s + 1.31·23-s − 0.0546·24-s − 1.60·26-s + 0.571·27-s − 0.293·28-s + 1.28·29-s − 0.946·31-s + 1.34·32-s + 0.342·34-s − 0.788·36-s − 1.10·37-s + 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 3 | \( 1 + 0.517T + 3T^{2} \) |
| 7 | \( 1 + 0.896T + 7T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 1.03T + 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 - 6.31T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 + 6.69T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 4.73T + 59T^{2} \) |
| 61 | \( 1 - 8.26T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 - 5.46T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 6.46T + 89T^{2} \) |
| 97 | \( 1 + 0.656T + 97T^{2} \) |
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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
−8.504897286190801967189280352922, −7.939004951388928204422637543348, −6.74778238496862198240670585147, −6.47230313360010601980553456623, −5.41969858233680054287916381094, −4.46338634088067092376584306036, −3.36274344519274170373440833788, −2.30322949296656629952665844367, −1.12615663508548652106420889042, 0,
1.12615663508548652106420889042, 2.30322949296656629952665844367, 3.36274344519274170373440833788, 4.46338634088067092376584306036, 5.41969858233680054287916381094, 6.47230313360010601980553456623, 6.74778238496862198240670585147, 7.939004951388928204422637543348, 8.504897286190801967189280352922