| L(s) = 1 | + (0.483 + 1.66i)3-s − i·5-s − 0.0981i·7-s + (−2.53 + 1.60i)9-s − 2.90·11-s + 13-s + (1.66 − 0.483i)15-s − 0.534i·17-s − 3.49i·19-s + (0.163 − 0.0475i)21-s − 1.68·23-s − 25-s + (−3.90 − 3.43i)27-s − 5.89i·29-s + 2.61i·31-s + ⋯ |
| L(s) = 1 | + (0.279 + 0.960i)3-s − 0.447i·5-s − 0.0371i·7-s + (−0.843 + 0.536i)9-s − 0.875·11-s + 0.277·13-s + (0.429 − 0.124i)15-s − 0.129i·17-s − 0.801i·19-s + (0.0356 − 0.0103i)21-s − 0.352·23-s − 0.200·25-s + (−0.750 − 0.660i)27-s − 1.09i·29-s + 0.470i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375862180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375862180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.483 - 1.66i)T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 0.0981iT - 7T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 17 | \( 1 + 0.534iT - 17T^{2} \) |
| 19 | \( 1 + 3.49iT - 19T^{2} \) |
| 23 | \( 1 + 1.68T + 23T^{2} \) |
| 29 | \( 1 + 5.89iT - 29T^{2} \) |
| 31 | \( 1 - 2.61iT - 31T^{2} \) |
| 37 | \( 1 - 9.99T + 37T^{2} \) |
| 41 | \( 1 + 8.69iT - 41T^{2} \) |
| 43 | \( 1 + 2.62iT - 43T^{2} \) |
| 47 | \( 1 + 0.623T + 47T^{2} \) |
| 53 | \( 1 + 1.38iT - 53T^{2} \) |
| 59 | \( 1 - 1.51T + 59T^{2} \) |
| 61 | \( 1 + 0.617T + 61T^{2} \) |
| 67 | \( 1 - 1.41iT - 67T^{2} \) |
| 71 | \( 1 + 7.58T + 71T^{2} \) |
| 73 | \( 1 + 7.02T + 73T^{2} \) |
| 79 | \( 1 + 0.0680iT - 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 14.5iT - 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.726993797703578711253619486984, −7.989801028900106761781043947047, −7.30868700412313919775334904309, −6.08257322883250074749628495483, −5.44460663644189931080158430989, −4.64441985761967610199991043716, −4.01920031495411680955500315177, −2.97792820516369128950895169251, −2.17221700424471946735075664046, −0.44034015884500696382312438731,
1.12118227896764781148240796504, 2.24576380614365456011980674362, 2.98092754740584002016985244017, 3.89175951938429902353982591662, 5.08556372190374738041061252631, 6.02556504793285761946297350581, 6.43531236724296051571970226118, 7.55332092231908247101901351158, 7.79598659825409209532309182318, 8.611468250688982242963297887191
