| L(s) = 1 | + (−1.49 − 1.66i)5-s − 1.43i·7-s + 0.0984i·11-s + 1.92·13-s + 2.03i·17-s − 0.999i·19-s + 4.30i·23-s + (−0.525 + 4.97i)25-s + 3.56i·29-s − 1.42·31-s + (−2.38 + 2.14i)35-s + 8.27·37-s + 5.11·41-s + 5.50·43-s + 9.00i·47-s + ⋯ |
| L(s) = 1 | + (−0.668 − 0.743i)5-s − 0.542i·7-s + 0.0296i·11-s + 0.533·13-s + 0.494i·17-s − 0.229i·19-s + 0.897i·23-s + (−0.105 + 0.994i)25-s + 0.662i·29-s − 0.255·31-s + (−0.403 + 0.362i)35-s + 1.36·37-s + 0.799·41-s + 0.839·43-s + 1.31i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.617346761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.617346761\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.49 + 1.66i)T \) |
| good | 7 | \( 1 + 1.43iT - 7T^{2} \) |
| 11 | \( 1 - 0.0984iT - 11T^{2} \) |
| 13 | \( 1 - 1.92T + 13T^{2} \) |
| 17 | \( 1 - 2.03iT - 17T^{2} \) |
| 19 | \( 1 + 0.999iT - 19T^{2} \) |
| 23 | \( 1 - 4.30iT - 23T^{2} \) |
| 29 | \( 1 - 3.56iT - 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 - 8.27T + 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 - 5.50T + 43T^{2} \) |
| 47 | \( 1 - 9.00iT - 47T^{2} \) |
| 53 | \( 1 - 4.93T + 53T^{2} \) |
| 59 | \( 1 + 6.27iT - 59T^{2} \) |
| 61 | \( 1 + 12.8iT - 61T^{2} \) |
| 67 | \( 1 + 1.89T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 5.62iT - 73T^{2} \) |
| 79 | \( 1 + 2.48T + 79T^{2} \) |
| 83 | \( 1 + 2.26T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 14.7iT - 97T^{2} \) |
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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
−8.111987284590989689209161492828, −7.76798389112437665546127906812, −6.97929713388460763332912992290, −6.06158347244550990353712446202, −5.32728395593143025178294220366, −4.39504572582607645374699505624, −3.90661065532186911945088423826, −2.99906455615988935632760264117, −1.62643573311265266701706586719, −0.71572168017621088893666579715,
0.74705971641215113119419682716, 2.30465624955305066303354146146, 2.89728018499786553744896948662, 3.94077982979509551382483713848, 4.49766129435663359332531976323, 5.72947685366353445785988899656, 6.14637944791473272228044495357, 7.14292421023097488916400584983, 7.58626907201970600317918220428, 8.518869332113574128215776323321
