| L(s) = 1 | + 2-s + 4-s + (0.648 + 2.56i)7-s + 8-s + 1.29i·11-s − 3.13·13-s + (0.648 + 2.56i)14-s + 16-s + 5.53i·17-s − 7.37i·19-s + 1.29i·22-s + 1.83·23-s − 3.13·26-s + (0.648 + 2.56i)28-s + 1.83i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.245 + 0.969i)7-s + 0.353·8-s + 0.390i·11-s − 0.868·13-s + (0.173 + 0.685i)14-s + 0.250·16-s + 1.34i·17-s − 1.69i·19-s + 0.276i·22-s + 0.382·23-s − 0.613·26-s + (0.122 + 0.484i)28-s + 0.340i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0955 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0955 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.432385317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.432385317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.648 - 2.56i)T \) |
| good | 11 | \( 1 - 1.29iT - 11T^{2} \) |
| 13 | \( 1 + 3.13T + 13T^{2} \) |
| 17 | \( 1 - 5.53iT - 17T^{2} \) |
| 19 | \( 1 + 7.37iT - 19T^{2} \) |
| 23 | \( 1 - 1.83T + 23T^{2} \) |
| 29 | \( 1 - 1.83iT - 29T^{2} \) |
| 31 | \( 1 - 10.4iT - 31T^{2} \) |
| 37 | \( 1 - 10.6iT - 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 - 3.53iT - 43T^{2} \) |
| 47 | \( 1 + 10.7iT - 47T^{2} \) |
| 53 | \( 1 - 4.42T + 53T^{2} \) |
| 59 | \( 1 - 7.18T + 59T^{2} \) |
| 61 | \( 1 + 4.88iT - 61T^{2} \) |
| 67 | \( 1 - 9.79iT - 67T^{2} \) |
| 71 | \( 1 - 7.37iT - 71T^{2} \) |
| 73 | \( 1 + 3.40T + 73T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 - 6.26iT - 83T^{2} \) |
| 89 | \( 1 + 7.94T + 89T^{2} \) |
| 97 | \( 1 + 8.09T + 97T^{2} \) |
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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
−8.632084408201940679830593753236, −8.337639844839377222739961624631, −6.94396833885385829294129704589, −6.83481319028271357075636483866, −5.61459687910061527218558287040, −5.05069047959617342060514437252, −4.41577781559922645496494518742, −3.20587245135583765350188560769, −2.50788920139638131762570340519, −1.51468298139453488031707597266,
0.56035713599072284673937367531, 1.92438458810233149404694516545, 2.93556186395220294260344648642, 3.90760292369328577936566438318, 4.47857167567494955021794108951, 5.43399579006752365645245882127, 6.03890572602552242422539156764, 7.15411943736969473019869507468, 7.49548744485782349220569061004, 8.237966170505910174196266981578
