| L(s) = 1 | − 1.27·5-s − 4.72·7-s − 7.27·11-s + 4.30i·13-s + 20.3·17-s + (7.54 − 17.4i)19-s + 5.45·23-s − 23.3·25-s + 8.60i·29-s + 20.0i·31-s + 6.02·35-s − 40.6i·37-s − 31.2i·41-s − 65.1·43-s + 55.4·47-s + ⋯ |
| L(s) = 1 | − 0.254·5-s − 0.675·7-s − 0.661·11-s + 0.330i·13-s + 1.19·17-s + (0.397 − 0.917i)19-s + 0.236·23-s − 0.934·25-s + 0.296i·29-s + 0.647i·31-s + 0.172·35-s − 1.09i·37-s − 0.763i·41-s − 1.51·43-s + 1.18·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.177187777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.177187777\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-7.54 + 17.4i)T \) |
| good | 5 | \( 1 + 1.27T + 25T^{2} \) |
| 7 | \( 1 + 4.72T + 49T^{2} \) |
| 11 | \( 1 + 7.27T + 121T^{2} \) |
| 13 | \( 1 - 4.30iT - 169T^{2} \) |
| 17 | \( 1 - 20.3T + 289T^{2} \) |
| 23 | \( 1 - 5.45T + 529T^{2} \) |
| 29 | \( 1 - 8.60iT - 841T^{2} \) |
| 31 | \( 1 - 20.0iT - 961T^{2} \) |
| 37 | \( 1 + 40.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 31.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 65.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 55.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 78.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 69.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 6.17T + 3.72e3T^{2} \) |
| 67 | \( 1 - 123. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 3.11iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 33.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 87.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 121.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 69.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 111. iT - 9.40e3T^{2} \) |
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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
−8.850300288657049616425516443238, −7.948166204846253142839815562797, −7.28946338038731159378629299154, −6.60764760434134270690516554568, −5.59685589848387407461644530126, −5.03486933194356508520482830013, −3.85834801496061482588278984212, −3.19198139028928648008705841987, −2.19284823676519757991527080399, −0.807235942050273117973538923413,
0.36159784590033115359459843080, 1.65572040906560433503066010716, 2.97846614283934586805442402227, 3.49660481014060633893714910031, 4.57319753666123868508719120765, 5.54339152044358087029621923722, 6.08836386444857457706480786143, 7.05703904630177754701991498384, 7.985090201294193751496099922355, 8.147477884777442769328866618186
