| L(s) = 1 | + (−2.14 + 0.620i)5-s + 1.65i·7-s − 4.14i·11-s + 5.20i·13-s + 2.42·17-s − 3.22i·19-s + 8.09i·23-s + (4.22 − 2.66i)25-s + (−5.37 + 0.292i)29-s − 8.43i·31-s + (−1.02 − 3.54i)35-s − 9.87·37-s + 10.7i·41-s − 2.24·43-s + 5.20·47-s + ⋯ |
| L(s) = 1 | + (−0.960 + 0.277i)5-s + 0.624i·7-s − 1.25i·11-s + 1.44i·13-s + 0.587·17-s − 0.740i·19-s + 1.68i·23-s + (0.845 − 0.533i)25-s + (−0.998 + 0.0542i)29-s − 1.51i·31-s + (−0.173 − 0.599i)35-s − 1.62·37-s + 1.67i·41-s − 0.341·43-s + 0.759·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1392880791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1392880791\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.14 - 0.620i)T \) |
| 29 | \( 1 + (5.37 - 0.292i)T \) |
| good | 7 | \( 1 - 1.65iT - 7T^{2} \) |
| 11 | \( 1 + 4.14iT - 11T^{2} \) |
| 13 | \( 1 - 5.20iT - 13T^{2} \) |
| 17 | \( 1 - 2.42T + 17T^{2} \) |
| 19 | \( 1 + 3.22iT - 19T^{2} \) |
| 23 | \( 1 - 8.09iT - 23T^{2} \) |
| 31 | \( 1 + 8.43iT - 31T^{2} \) |
| 37 | \( 1 + 9.87T + 37T^{2} \) |
| 41 | \( 1 - 10.7iT - 41T^{2} \) |
| 43 | \( 1 + 2.24T + 43T^{2} \) |
| 47 | \( 1 - 5.20T + 47T^{2} \) |
| 53 | \( 1 - 4.67iT - 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 - 10.1iT - 61T^{2} \) |
| 67 | \( 1 + 1.51iT - 67T^{2} \) |
| 71 | \( 1 - 7.48T + 71T^{2} \) |
| 73 | \( 1 - 7.38T + 73T^{2} \) |
| 79 | \( 1 + 5.98iT - 79T^{2} \) |
| 83 | \( 1 + 5.74iT - 83T^{2} \) |
| 89 | \( 1 + 16.2iT - 89T^{2} \) |
| 97 | \( 1 - 4.90T + 97T^{2} \) |
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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
−8.726951255819679862503938670672, −7.74563696439392477138896459335, −7.34791849675123345590326442619, −6.42081436263675471571975953989, −5.77386802042044168766216545622, −4.96671437481827405453188238826, −3.98083221372595723505037767354, −3.44228632238062867929948089128, −2.55332059730179480634098287716, −1.37865505096975023422311865291,
0.04159787598220656364594882727, 1.12760405110050423856871020320, 2.33901762585304742042454920435, 3.53716144325895339744642936626, 3.88860760434203111819328972299, 4.99327751235199923186926575952, 5.32301528715297670646218837901, 6.61637698088803811296569205740, 7.19468320028616740790389406993, 7.80158866259272508486498650557
