L(s) = 1 | − 5-s + 3·7-s − 3·11-s + 4·17-s + 6·19-s + 6·23-s − 4·25-s − 2·29-s + 9·31-s − 3·35-s − 2·37-s − 10·41-s + 6·43-s + 6·47-s + 2·49-s + 13·53-s + 3·55-s − 12·59-s + 8·61-s + 6·67-s + 12·71-s + 9·73-s − 9·77-s − 3·83-s − 4·85-s + 14·89-s − 6·95-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.13·7-s − 0.904·11-s + 0.970·17-s + 1.37·19-s + 1.25·23-s − 4/5·25-s − 0.371·29-s + 1.61·31-s − 0.507·35-s − 0.328·37-s − 1.56·41-s + 0.914·43-s + 0.875·47-s + 2/7·49-s + 1.78·53-s + 0.404·55-s − 1.56·59-s + 1.02·61-s + 0.733·67-s + 1.42·71-s + 1.05·73-s − 1.02·77-s − 0.329·83-s − 0.433·85-s + 1.48·89-s − 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.643655999\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.643655999\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−10.24151483342147504043812250434, −9.344827270962992036294829473440, −8.180464945333468393072503118033, −7.82476086711398995912338456757, −6.93499396476405803707004808409, −5.45678967002093979727974577207, −5.03134570361553185426866134499, −3.77979388504550471995644973896, −2.63884396406011957309091091690, −1.11568290382473683806344984693,
1.11568290382473683806344984693, 2.63884396406011957309091091690, 3.77979388504550471995644973896, 5.03134570361553185426866134499, 5.45678967002093979727974577207, 6.93499396476405803707004808409, 7.82476086711398995912338456757, 8.180464945333468393072503118033, 9.344827270962992036294829473440, 10.24151483342147504043812250434