L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 1.67·11-s − 12-s − 4.48·13-s + 16-s − 7.61·17-s − 18-s + 4.48·19-s − 1.67·22-s + 0.482·23-s + 24-s + 4.48·26-s − 27-s + 1.19·29-s − 3.48·31-s − 32-s − 1.67·33-s + 7.61·34-s + 36-s + 0.482·37-s − 4.48·38-s + 4.48·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.505·11-s − 0.288·12-s − 1.24·13-s + 0.250·16-s − 1.84·17-s − 0.235·18-s + 1.02·19-s − 0.357·22-s + 0.100·23-s + 0.204·24-s + 0.879·26-s − 0.192·27-s + 0.221·29-s − 0.625·31-s − 0.176·32-s − 0.291·33-s + 1.30·34-s + 0.166·36-s + 0.0793·37-s − 0.727·38-s + 0.717·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7247989151\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7247989151\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 4.48T + 13T^{2} \) |
| 17 | \( 1 + 7.61T + 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 - 0.482T + 23T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 31 | \( 1 + 3.48T + 31T^{2} \) |
| 37 | \( 1 - 0.482T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 9.57T + 53T^{2} \) |
| 59 | \( 1 - 5.77T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 3.35T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 + 1.87T + 83T^{2} \) |
| 89 | \( 1 + 3.35T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−7.79162077789219675945921167937, −7.15830929210379213454490699951, −6.71297822941553471835126847894, −5.95409394286518012476969506343, −5.06944160856705618755249813651, −4.51128520063748240406843190568, −3.48490080427250521537798959424, −2.46827835199970228117572843532, −1.68817005675642788485807164296, −0.48561816343868350318216596720,
0.48561816343868350318216596720, 1.68817005675642788485807164296, 2.46827835199970228117572843532, 3.48490080427250521537798959424, 4.51128520063748240406843190568, 5.06944160856705618755249813651, 5.95409394286518012476969506343, 6.71297822941553471835126847894, 7.15830929210379213454490699951, 7.79162077789219675945921167937