| L(s) = 1 | − 0.784i·2-s − i·3-s + 1.38·4-s + (−1.83 + 1.27i)5-s − 0.784·6-s + 1.51i·7-s − 2.65i·8-s − 9-s + (0.996 + 1.44i)10-s − 1.38i·12-s − 0.466i·13-s + 1.19·14-s + (1.27 + 1.83i)15-s + 0.688·16-s − 1.69i·17-s + 0.784i·18-s + ⋯ |
| L(s) = 1 | − 0.554i·2-s − 0.577i·3-s + 0.692·4-s + (−0.822 + 0.568i)5-s − 0.320·6-s + 0.573i·7-s − 0.938i·8-s − 0.333·9-s + (0.315 + 0.456i)10-s − 0.399i·12-s − 0.129i·13-s + 0.318·14-s + (0.328 + 0.474i)15-s + 0.172·16-s − 0.411i·17-s + 0.184i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.875169960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875169960\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.83 - 1.27i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.784iT - 2T^{2} \) |
| 7 | \( 1 - 1.51iT - 7T^{2} \) |
| 13 | \( 1 + 0.466iT - 13T^{2} \) |
| 17 | \( 1 + 1.69iT - 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 - 7.68iT - 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 - 3.05iT - 37T^{2} \) |
| 41 | \( 1 - 1.35T + 41T^{2} \) |
| 43 | \( 1 + 12.0iT - 43T^{2} \) |
| 47 | \( 1 + 2.55iT - 47T^{2} \) |
| 53 | \( 1 - 8.01iT - 53T^{2} \) |
| 59 | \( 1 + 7.54T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 6.89iT - 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 2.26iT - 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 11.7iT - 83T^{2} \) |
| 89 | \( 1 + 1.47T + 89T^{2} \) |
| 97 | \( 1 - 14.2iT - 97T^{2} \) |
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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
−9.241399463966681174042197516664, −8.163890764622222362509338949510, −7.45291886260043200184601115362, −6.96809143246441919824827906698, −6.05902315233436459593687928620, −5.15996292694342926063898704154, −3.71513520359334632518292216149, −3.06290027324200978499144705422, −2.21411326269331223349780775945, −0.927190098842522668300772700259,
0.979941528984422150761567468024, 2.60583981348942736384877546649, 3.64013218727027691232513147881, 4.55197780201749598772956694765, 5.25430650284853570310739179556, 6.30788092632625171544792609724, 7.02159839064529262982776361872, 7.938014390865948331313850804767, 8.300799794389129368940807526384, 9.302795218258283029437610923822
