| L(s) = 1 | − 5.67·5-s + 33.0·7-s − 34.6·11-s + 82.2·13-s − 97.8·17-s − 55.8·19-s + 130.·23-s − 92.7·25-s + 147.·29-s + 101.·31-s − 187.·35-s − 184.·37-s + 237.·41-s + 199.·43-s + 334.·47-s + 752.·49-s + 102.·53-s + 196.·55-s − 105.·59-s − 717.·61-s − 466.·65-s + 316.·67-s + 800.·71-s − 301.·73-s − 1.14e3·77-s − 42.8·79-s − 1.23e3·83-s + ⋯ |
| L(s) = 1 | − 0.507·5-s + 1.78·7-s − 0.949·11-s + 1.75·13-s − 1.39·17-s − 0.674·19-s + 1.18·23-s − 0.742·25-s + 0.944·29-s + 0.586·31-s − 0.907·35-s − 0.819·37-s + 0.904·41-s + 0.708·43-s + 1.03·47-s + 2.19·49-s + 0.264·53-s + 0.481·55-s − 0.233·59-s − 1.50·61-s − 0.891·65-s + 0.577·67-s + 1.33·71-s − 0.482·73-s − 1.69·77-s − 0.0610·79-s − 1.63·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.553041892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.553041892\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.67T + 125T^{2} \) |
| 7 | \( 1 - 33.0T + 343T^{2} \) |
| 11 | \( 1 + 34.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 82.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 97.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 55.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 147.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 184.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 237.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 199.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 334.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 102.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 105.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 717.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 316.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 800.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 301.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 42.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 505.T + 9.12e5T^{2} \) |
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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
−8.502923552331907051036739629986, −8.069544055450568733918610559505, −7.26758768023207034050902992279, −6.30235480643947501023359244012, −5.40481355202480006302005631326, −4.55085966871602950544140674522, −4.04080881788574518486205223549, −2.71740994443654721742506875690, −1.74970668690922792386057788012, −0.74043429935996618801519879959,
0.74043429935996618801519879959, 1.74970668690922792386057788012, 2.71740994443654721742506875690, 4.04080881788574518486205223549, 4.55085966871602950544140674522, 5.40481355202480006302005631326, 6.30235480643947501023359244012, 7.26758768023207034050902992279, 8.069544055450568733918610559505, 8.502923552331907051036739629986
