| L(s) = 1 | + 0.353·2-s − 0.847·3-s − 1.87·4-s − 5-s − 0.299·6-s + 7-s − 1.36·8-s − 2.28·9-s − 0.353·10-s + 5.72·11-s + 1.58·12-s − 1.70·13-s + 0.353·14-s + 0.847·15-s + 3.26·16-s − 17-s − 0.806·18-s + 4.42·19-s + 1.87·20-s − 0.847·21-s + 2.02·22-s + 6.73·23-s + 1.15·24-s + 25-s − 0.600·26-s + 4.47·27-s − 1.87·28-s + ⋯ |
| L(s) = 1 | + 0.249·2-s − 0.489·3-s − 0.937·4-s − 0.447·5-s − 0.122·6-s + 0.377·7-s − 0.484·8-s − 0.760·9-s − 0.111·10-s + 1.72·11-s + 0.458·12-s − 0.471·13-s + 0.0944·14-s + 0.218·15-s + 0.816·16-s − 0.242·17-s − 0.190·18-s + 1.01·19-s + 0.419·20-s − 0.184·21-s + 0.430·22-s + 1.40·23-s + 0.236·24-s + 0.200·25-s − 0.117·26-s + 0.861·27-s − 0.354·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.057026250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057026250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 0.353T + 2T^{2} \) |
| 3 | \( 1 + 0.847T + 3T^{2} \) |
| 11 | \( 1 - 5.72T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 19 | \( 1 - 4.42T + 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 + 4.86T + 29T^{2} \) |
| 31 | \( 1 - 5.00T + 31T^{2} \) |
| 37 | \( 1 - 6.90T + 37T^{2} \) |
| 41 | \( 1 - 2.29T + 41T^{2} \) |
| 43 | \( 1 - 0.713T + 43T^{2} \) |
| 47 | \( 1 + 4.87T + 47T^{2} \) |
| 53 | \( 1 - 6.54T + 53T^{2} \) |
| 59 | \( 1 - 6.98T + 59T^{2} \) |
| 61 | \( 1 + 9.63T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 3.18T + 71T^{2} \) |
| 73 | \( 1 - 6.60T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 7.30T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−10.91663005362861344715796522106, −9.553519568532240214822171120263, −9.055181990251289222636087733023, −8.144780303075742819701977066710, −7.02435682491380330789149725431, −5.96665366816176924045228036993, −5.03105084863530835049732712644, −4.20672486677157284530955045698, −3.11690228917585286589374089600, −0.932778701362221718879136132942,
0.932778701362221718879136132942, 3.11690228917585286589374089600, 4.20672486677157284530955045698, 5.03105084863530835049732712644, 5.96665366816176924045228036993, 7.02435682491380330789149725431, 8.144780303075742819701977066710, 9.055181990251289222636087733023, 9.553519568532240214822171120263, 10.91663005362861344715796522106
