| L(s) = 1 | + 29.2·3-s + 7.45·5-s + 613.·9-s + 447.·11-s − 383.·13-s + 218.·15-s + 1.87e3·17-s − 7.59·19-s + 2.69e3·23-s − 3.06e3·25-s + 1.08e4·27-s − 8.77e3·29-s − 7.54e3·31-s + 1.31e4·33-s + 6.51e3·37-s − 1.12e4·39-s + 1.79e4·41-s + 1.35e3·43-s + 4.57e3·45-s − 1.82e3·47-s + 5.48e4·51-s + 3.58e4·53-s + 3.33e3·55-s − 222.·57-s + 2.36e4·59-s − 2.70e3·61-s − 2.85e3·65-s + ⋯ |
| L(s) = 1 | + 1.87·3-s + 0.133·5-s + 2.52·9-s + 1.11·11-s − 0.628·13-s + 0.250·15-s + 1.57·17-s − 0.00482·19-s + 1.06·23-s − 0.982·25-s + 2.86·27-s − 1.93·29-s − 1.41·31-s + 2.09·33-s + 0.782·37-s − 1.18·39-s + 1.67·41-s + 0.111·43-s + 0.337·45-s − 0.120·47-s + 2.95·51-s + 1.75·53-s + 0.148·55-s − 0.00905·57-s + 0.883·59-s − 0.0929·61-s − 0.0839·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.108671473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.108671473\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 29.2T + 243T^{2} \) |
| 5 | \( 1 - 7.45T + 3.12e3T^{2} \) |
| 11 | \( 1 - 447.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 383.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.87e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 7.59T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.54e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.79e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.82e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.58e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.70e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.37e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.44e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.18e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.67e4T + 8.58e9T^{2} \) |
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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
−10.03048375139327553972065022417, −9.400577064458578367530415869204, −8.839989070130205679577015125232, −7.59801312555641139827744200269, −7.27622235673546989176430846720, −5.66506778379543607395300923816, −4.11649368469342892697874876522, −3.42159025065552354856990024611, −2.27538717395031656058734955269, −1.23751684471206114363040509807,
1.23751684471206114363040509807, 2.27538717395031656058734955269, 3.42159025065552354856990024611, 4.11649368469342892697874876522, 5.66506778379543607395300923816, 7.27622235673546989176430846720, 7.59801312555641139827744200269, 8.839989070130205679577015125232, 9.400577064458578367530415869204, 10.03048375139327553972065022417
