| L(s) = 1 | + 3.46i·5-s + 3.46i·11-s + 1.73i·13-s − 5·19-s + 6.92i·23-s − 6.99·25-s + 5·31-s − 11·37-s − 3.46i·41-s − 8.66i·43-s − 6·47-s − 12·53-s − 11.9·55-s + 12·59-s + 13.8i·61-s + ⋯ |
| L(s) = 1 | + 1.54i·5-s + 1.04i·11-s + 0.480i·13-s − 1.14·19-s + 1.44i·23-s − 1.39·25-s + 0.898·31-s − 1.80·37-s − 0.541i·41-s − 1.32i·43-s − 0.875·47-s − 1.64·53-s − 1.61·55-s + 1.56·59-s + 1.77i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7650527888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7650527888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 8.66iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 8.66iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 5.19iT - 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 - 18T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−8.292580982209459627763642309998, −7.38685088385963137598086254672, −7.01209360470179645598128247662, −6.48461725564169419431711418332, −5.67227757354180340637167236313, −4.77491474989312996063551890508, −3.90920015233321817057365775941, −3.28198743354045338649249074554, −2.31488205253455149904513573313, −1.73013951865700231426939236081,
0.19705900936285821637887050259, 1.01635087258530469867790995319, 2.04489505835585470529200096226, 3.11873914347474651380903058748, 3.97286534375333830507953429899, 4.88863348474614068894793699188, 5.09657336443332539342772936812, 6.28279333165806120222328903693, 6.47637450865897529669155328298, 7.88844956542213939533220870752
