L(s) = 1 | − 15.4·3-s − 58.0·5-s + 210.·7-s − 5.16·9-s − 527.·11-s − 92.3·13-s + 895.·15-s + 1.79e3·17-s − 1.63e3·19-s − 3.24e3·21-s + 2.76e3·23-s + 248.·25-s + 3.82e3·27-s + 841·29-s − 689.·31-s + 8.13e3·33-s − 1.22e4·35-s − 1.27e3·37-s + 1.42e3·39-s + 1.80e4·41-s + 6.41e3·43-s + 300.·45-s − 2.06e3·47-s + 2.74e4·49-s − 2.76e4·51-s − 2.87e4·53-s + 3.06e4·55-s + ⋯ |
L(s) = 1 | − 0.989·3-s − 1.03·5-s + 1.62·7-s − 0.0212·9-s − 1.31·11-s − 0.151·13-s + 1.02·15-s + 1.50·17-s − 1.04·19-s − 1.60·21-s + 1.09·23-s + 0.0795·25-s + 1.01·27-s + 0.185·29-s − 0.128·31-s + 1.29·33-s − 1.68·35-s − 0.153·37-s + 0.149·39-s + 1.67·41-s + 0.529·43-s + 0.0220·45-s − 0.136·47-s + 1.63·49-s − 1.48·51-s − 1.40·53-s + 1.36·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 - 841T \) |
good | 3 | \( 1 + 15.4T + 243T^{2} \) |
| 5 | \( 1 + 58.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 210.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 527.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 92.3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.79e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.63e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.76e3T + 6.43e6T^{2} \) |
| 31 | \( 1 + 689.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.27e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.80e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.41e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.06e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.87e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.47e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.80e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.17e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.62e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.78e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.03e4T + 8.58e9T^{2} \) |
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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
−10.15738476491355275765372744015, −8.537608187844706813272912577273, −7.947806410578072927912898390395, −7.23120635243881306630267199628, −5.71173496969399945643773715946, −5.09206762075595970478757957714, −4.22825068638107652843352004166, −2.70431850327739513327141225295, −1.11600986005915270838000614450, 0,
1.11600986005915270838000614450, 2.70431850327739513327141225295, 4.22825068638107652843352004166, 5.09206762075595970478757957714, 5.71173496969399945643773715946, 7.23120635243881306630267199628, 7.947806410578072927912898390395, 8.537608187844706813272912577273, 10.15738476491355275765372744015