| L(s) = 1 | + 4.68·5-s + (−7.47 + 16.9i)7-s + 18.6i·11-s + 50.9i·13-s + 112.·17-s + 111. i·19-s + 143. i·23-s − 103.·25-s + 238. i·29-s − 207. i·31-s + (−35.0 + 79.3i)35-s − 227.·37-s + 266.·41-s + 340.·43-s − 223.·47-s + ⋯ |
| L(s) = 1 | + 0.419·5-s + (−0.403 + 0.914i)7-s + 0.510i·11-s + 1.08i·13-s + 1.60·17-s + 1.34i·19-s + 1.30i·23-s − 0.824·25-s + 1.52i·29-s − 1.20i·31-s + (−0.169 + 0.383i)35-s − 1.01·37-s + 1.01·41-s + 1.20·43-s − 0.695·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.403i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.750455565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.750455565\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (7.47 - 16.9i)T \) |
| good | 5 | \( 1 - 4.68T + 125T^{2} \) |
| 11 | \( 1 - 18.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 50.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 143. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 238. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 207. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 227.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 340.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 223.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 547. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 87.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 360. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 744.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 135. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 467. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 384.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3iT - 9.12e5T^{2} \) |
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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
−9.247883749295830997088141483392, −8.203624423347126633380757974401, −7.50134762922745696639970057040, −6.62345285390259798968733258663, −5.66187767844535212972786165662, −5.41427891284524260090083918466, −4.03329606250613309549501549598, −3.30003096820497479293494991851, −2.10326079610766884887006779786, −1.42176721786845246598758919322,
0.37592212268056695786765103396, 1.05558682142015035113228827675, 2.56464926202511584319448404650, 3.30276703538097070804884872210, 4.24617767678226555985256510886, 5.25123680851189913936577302650, 5.98602642656646800132369616900, 6.75013720763002182220221173479, 7.66123320950068055846790980958, 8.174673279696746488617431049277
