L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s + 5-s − 1.00·6-s + 7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)10-s + (0.707 + 0.707i)12-s + (−0.707 + 0.707i)13-s + (−0.707 − 0.707i)14-s + (0.707 − 0.707i)15-s − 1.00·16-s − i·17-s + (−0.707 − 0.707i)19-s + 1.00i·20-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s + 5-s − 1.00·6-s + 7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)10-s + (0.707 + 0.707i)12-s + (−0.707 + 0.707i)13-s + (−0.707 − 0.707i)14-s + (0.707 − 0.707i)15-s − 1.00·16-s − i·17-s + (−0.707 − 0.707i)19-s + 1.00i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.238626370\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.238626370\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 17 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + 2iT - T^{2} \) |
| 53 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 61 | \( 1 + (1 - i)T - iT^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−9.480217128296803467047098736327, −8.644135484760234173464514621033, −8.222107689824568944525921702630, −7.19144120246753687900878863812, −6.74958525031718999777366570596, −5.19049985355607847629695486041, −4.42826524233722073389210582820, −2.85258647534314414768309836966, −2.19243334271598069788543803958, −1.48100057894374379630253546678,
1.58677514076755699922724424496, 2.54865463745511774987780068894, 4.07927362806563423011033827363, 4.94165594099760477761444161881, 5.83968617075900742335883966206, 6.49972656187667059159101531694, 7.75624483145874433734264790615, 8.322650848275652444843595489597, 8.882224936606427437036550063301, 9.915707352400091226204629461296