| L(s) = 1 | + (2 − 1.73i)7-s − 3.46i·11-s − 6.92i·17-s − 4·19-s + 3.46i·23-s + 5·25-s − 6·29-s − 4·31-s + 2·37-s − 6.92i·41-s − 3.46i·43-s + (1.00 − 6.92i)49-s − 6·53-s − 12·59-s + 13.8i·61-s + ⋯ |
| L(s) = 1 | + (0.755 − 0.654i)7-s − 1.04i·11-s − 1.68i·17-s − 0.917·19-s + 0.722i·23-s + 25-s − 1.11·29-s − 0.718·31-s + 0.328·37-s − 1.08i·41-s − 0.528i·43-s + (0.142 − 0.989i)49-s − 0.824·53-s − 1.56·59-s + 1.77i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.262323931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.262323931\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 + 1.73i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6.92iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.140236842863043939592997127714, −7.36596129621046186560102438549, −6.90942699995464603918491130614, −5.80215894323858054924559630081, −5.20186014165233210085862576695, −4.36361081922535137529644733429, −3.54179344349884505746062420163, −2.62137315460838839491748861683, −1.45914790337154626334973100742, −0.34813396758386217322458515654,
1.58150446048113745780406392725, 2.13203154132026791604788157824, 3.29200290169809319248342112313, 4.41821273692427832149287134388, 4.77859774358522471463383159836, 5.88882031785211984023459342865, 6.39055698309630441475480556291, 7.36692360576498340970342407118, 8.065586621456287072468669383376, 8.650160060374412327563999776077
