| L(s) = 1 | + 3-s − 3·5-s + 4·7-s + 9-s − 13-s − 3·15-s − 2·17-s + 4·21-s + 4·25-s + 27-s − 29-s − 8·31-s − 12·35-s − 6·37-s − 39-s − 5·41-s + 12·43-s − 3·45-s + 12·47-s + 9·49-s − 2·51-s + 9·53-s + 8·59-s − 11·61-s + 4·63-s + 3·65-s − 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.774·15-s − 0.485·17-s + 0.872·21-s + 4/5·25-s + 0.192·27-s − 0.185·29-s − 1.43·31-s − 2.02·35-s − 0.986·37-s − 0.160·39-s − 0.780·41-s + 1.82·43-s − 0.447·45-s + 1.75·47-s + 9/7·49-s − 0.280·51-s + 1.23·53-s + 1.04·59-s − 1.40·61-s + 0.503·63-s + 0.372·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−15.40071468980695, −15.10415099625356, −14.67196158411438, −14.08224958676795, −13.65464613749870, −12.88764517223860, −12.22127316193771, −11.87973955569373, −11.34942650541755, −10.64906720431992, −10.53623253728547, −9.350039054432032, −8.831911871172777, −8.488952215236347, −7.768359751258034, −7.384367280524296, −7.102895240398673, −5.937154564360496, −5.240992253607260, −4.607198198408520, −4.052646973934307, −3.620843231583774, −2.623836491234191, −1.963518892832839, −1.116941615575991, 0,
1.116941615575991, 1.963518892832839, 2.623836491234191, 3.620843231583774, 4.052646973934307, 4.607198198408520, 5.240992253607260, 5.937154564360496, 7.102895240398673, 7.384367280524296, 7.768359751258034, 8.488952215236347, 8.831911871172777, 9.350039054432032, 10.53623253728547, 10.64906720431992, 11.34942650541755, 11.87973955569373, 12.22127316193771, 12.88764517223860, 13.65464613749870, 14.08224958676795, 14.67196158411438, 15.10415099625356, 15.40071468980695
