| L(s) = 1 | + i·3-s + (−0.594 − 2.15i)5-s − 4.92i·7-s − 9-s − 1.38·11-s − i·13-s + (2.15 − 0.594i)15-s + 0.195i·17-s + 4.92·21-s + 2.19i·23-s + (−4.29 + 2.56i)25-s − i·27-s − 7.49·29-s − 1.38i·33-s + (−10.6 + 2.92i)35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.265 − 0.964i)5-s − 1.86i·7-s − 0.333·9-s − 0.417·11-s − 0.277i·13-s + (0.556 − 0.153i)15-s + 0.0473i·17-s + 1.07·21-s + 0.457i·23-s + (−0.858 + 0.512i)25-s − 0.192i·27-s − 1.39·29-s − 0.240i·33-s + (−1.79 + 0.494i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6553344200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6553344200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.594 + 2.15i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 + 4.92iT - 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 17 | \( 1 - 0.195iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.19iT - 23T^{2} \) |
| 29 | \( 1 + 7.49T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.05iT - 37T^{2} \) |
| 41 | \( 1 + 2.11T + 41T^{2} \) |
| 43 | \( 1 + 3.49iT - 43T^{2} \) |
| 47 | \( 1 - 2.31iT - 47T^{2} \) |
| 53 | \( 1 - 6.05iT - 53T^{2} \) |
| 59 | \( 1 + 9.43T + 59T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 + 12.6iT - 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 4.73iT - 73T^{2} \) |
| 79 | \( 1 - 8.07T + 79T^{2} \) |
| 83 | \( 1 - 13.3iT - 83T^{2} \) |
| 89 | \( 1 + 3.02T + 89T^{2} \) |
| 97 | \( 1 - 6.01iT - 97T^{2} \) |
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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
−9.273831895498181575760700431621, −8.084174434591390361182344761893, −7.71670805861769610313592258061, −6.75126989178586260274899614547, −5.55532429237078362948135380097, −4.75043749347286715718374147780, −4.04707994755069372793524384088, −3.30994677112363163383904055973, −1.49630154741904672877877344022, −0.24881212121382622066677719239,
2.02336905634962973616370433155, 2.61589650855160782176645377454, 3.62211706043300704747487473997, 5.07504284907568751533120968006, 5.86914270302849143526352459960, 6.50082270628751814625291358022, 7.42726868854685323387569160570, 8.159069865780871951071979751688, 8.953637121148807609869121605570, 9.653078387186942830069685928210
