| L(s) = 1 | − 5.03·2-s − 8.47·3-s + 17.3·4-s − 0.885·5-s + 42.6·6-s − 46.9·8-s + 44.8·9-s + 4.45·10-s − 52.3·11-s − 146.·12-s + 8.06·13-s + 7.50·15-s + 97.5·16-s + 17·17-s − 225.·18-s + 66.5·19-s − 15.3·20-s + 263.·22-s + 180.·23-s + 397.·24-s − 124.·25-s − 40.5·26-s − 151.·27-s − 41.2·29-s − 37.7·30-s + 34.9·31-s − 115.·32-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 1.63·3-s + 2.16·4-s − 0.0792·5-s + 2.90·6-s − 2.07·8-s + 1.66·9-s + 0.140·10-s − 1.43·11-s − 3.53·12-s + 0.171·13-s + 0.129·15-s + 1.52·16-s + 0.242·17-s − 2.95·18-s + 0.803·19-s − 0.171·20-s + 2.55·22-s + 1.63·23-s + 3.38·24-s − 0.993·25-s − 0.305·26-s − 1.07·27-s − 0.264·29-s − 0.229·30-s + 0.202·31-s − 0.638·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2564375993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2564375993\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 5.03T + 8T^{2} \) |
| 3 | \( 1 + 8.47T + 27T^{2} \) |
| 5 | \( 1 + 0.885T + 125T^{2} \) |
| 11 | \( 1 + 52.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.06T + 2.19e3T^{2} \) |
| 19 | \( 1 - 66.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 34.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 130.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 329.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 678.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 340.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 15.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 670.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 193.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 379.T + 9.12e5T^{2} \) |
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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
−9.899925948486710916067759959567, −9.218355433454388926171800466699, −7.947146091553928964198897675405, −7.47445777971746959734942094536, −6.52696251430267962779673792006, −5.68733664890211892459609209476, −4.82248274769331596242166871067, −2.90582621941523246882679478281, −1.42704046574274373527607195824, −0.42577562269861993947339705204,
0.42577562269861993947339705204, 1.42704046574274373527607195824, 2.90582621941523246882679478281, 4.82248274769331596242166871067, 5.68733664890211892459609209476, 6.52696251430267962779673792006, 7.47445777971746959734942094536, 7.947146091553928964198897675405, 9.218355433454388926171800466699, 9.899925948486710916067759959567
