| L(s) = 1 | + (−0.707 − 0.707i)5-s − 1.41i·7-s + (−0.808 − 0.808i)11-s + (−0.749 + 0.749i)13-s − 5.97·17-s + (1.88 − 1.88i)19-s + 1.88i·23-s + 1.00i·25-s + (−5.88 + 5.88i)29-s − 1.61·31-s + (−1.00 + 1.00i)35-s + (3.69 + 3.69i)37-s + 8.77i·41-s + (0.744 + 0.744i)43-s + 13.5·47-s + ⋯ |
| L(s) = 1 | + (−0.316 − 0.316i)5-s − 0.534i·7-s + (−0.243 − 0.243i)11-s + (−0.207 + 0.207i)13-s − 1.44·17-s + (0.433 − 0.433i)19-s + 0.392i·23-s + 0.200i·25-s + (−1.09 + 1.09i)29-s − 0.289·31-s + (−0.169 + 0.169i)35-s + (0.607 + 0.607i)37-s + 1.36i·41-s + (0.113 + 0.113i)43-s + 1.97·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6996713979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6996713979\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 + 0.707i)T \) |
| good | 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + (0.808 + 0.808i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.749 - 0.749i)T - 13iT^{2} \) |
| 17 | \( 1 + 5.97T + 17T^{2} \) |
| 19 | \( 1 + (-1.88 + 1.88i)T - 19iT^{2} \) |
| 23 | \( 1 - 1.88iT - 23T^{2} \) |
| 29 | \( 1 + (5.88 - 5.88i)T - 29iT^{2} \) |
| 31 | \( 1 + 1.61T + 31T^{2} \) |
| 37 | \( 1 + (-3.69 - 3.69i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.77iT - 41T^{2} \) |
| 43 | \( 1 + (-0.744 - 0.744i)T + 43iT^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 + (0.863 + 0.863i)T + 53iT^{2} \) |
| 59 | \( 1 + (10.7 + 10.7i)T + 59iT^{2} \) |
| 61 | \( 1 + (9.05 - 9.05i)T - 61iT^{2} \) |
| 67 | \( 1 + (-2.94 + 2.94i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.78iT - 71T^{2} \) |
| 73 | \( 1 + 2.32iT - 73T^{2} \) |
| 79 | \( 1 - 4.27T + 79T^{2} \) |
| 83 | \( 1 + (3.11 - 3.11i)T - 83iT^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 3.72T + 97T^{2} \) |
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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
−9.121160619002339020058064442923, −8.183869712537988307561985525004, −7.44723144130986679962742718894, −6.85228502085014132708897477930, −5.92806229124839041397163422295, −4.97756503367273821376040676944, −4.32886382570303997500448367984, −3.44410290281083410825471861888, −2.38879302462772306209548752954, −1.14104102903563049681968498280,
0.23525554599325629966965285440, 2.00796516758080097919581730670, 2.69674689229131525770730573728, 3.86599360948902559940807984478, 4.54816045483342200531215270184, 5.62890785922134007027489981977, 6.16049459161057412367430862956, 7.29431623021330582463863311036, 7.59996155569478363240142233059, 8.715511251596435517178942072086
