| L(s) = 1 | + (−0.156 + 0.987i)2-s + (0.648 − 0.102i)3-s + (−0.951 − 0.309i)4-s + (1.87 + 1.22i)5-s + 0.656i·6-s + (0.676 + 1.32i)7-s + (0.453 − 0.891i)8-s + (−2.44 + 0.793i)9-s + (−1.49 + 1.65i)10-s + (1.63 + 0.530i)11-s + (−0.648 − 0.102i)12-s + (1.59 − 0.252i)13-s + (−1.41 + 0.460i)14-s + (1.34 + 0.599i)15-s + (0.809 + 0.587i)16-s + (0.365 − 0.717i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 + 0.698i)2-s + (0.374 − 0.0593i)3-s + (−0.475 − 0.154i)4-s + (0.838 + 0.545i)5-s + 0.268i·6-s + (0.255 + 0.501i)7-s + (0.160 − 0.315i)8-s + (−0.814 + 0.264i)9-s + (−0.473 + 0.524i)10-s + (0.492 + 0.159i)11-s + (−0.187 − 0.0296i)12-s + (0.441 − 0.0699i)13-s + (−0.378 + 0.122i)14-s + (0.346 + 0.154i)15-s + (0.202 + 0.146i)16-s + (0.0886 − 0.174i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11485 + 0.983747i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11485 + 0.983747i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.156 - 0.987i)T \) |
| 5 | \( 1 + (-1.87 - 1.22i)T \) |
| 31 | \( 1 + (-4.85 + 2.72i)T \) |
| good | 3 | \( 1 + (-0.648 + 0.102i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (-0.676 - 1.32i)T + (-4.11 + 5.66i)T^{2} \) |
| 11 | \( 1 + (-1.63 - 0.530i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.59 + 0.252i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.365 + 0.717i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.81 - 2.49i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.21 + 0.620i)T + (13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (3.25 - 2.36i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (5.96 + 5.96i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.954 - 0.693i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-0.876 + 5.53i)T + (-40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-2.74 + 0.433i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (8.48 + 4.32i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-6.98 + 9.61i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + 1.10iT - 61T^{2} \) |
| 67 | \( 1 + (-4.18 + 4.18i)T - 67iT^{2} \) |
| 71 | \( 1 + (-4.59 - 14.1i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (4.25 + 8.34i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (0.984 + 3.03i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.911 + 5.75i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-1.10 + 3.38i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (3.17 + 6.23i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.86991925674995416786866169411, −10.86802919353374501810147753939, −9.802665043049726091583743695440, −8.969121136252614815920639277121, −8.160852941580821932150370562237, −7.03082247938244605546594587402, −5.97753524464113734782855719249, −5.26863118229833681774168838703, −3.50776534296799874677354516033, −2.05102714001327095382756335365,
1.26919921904166329479434403423, 2.79477092152526637855923940953, 4.08757533452596213033702570857, 5.34076213022010212180842732192, 6.46086803625591951665641730115, 8.014379289835746443881137560394, 8.867369091159923199822779927518, 9.534531053040737029420459986323, 10.51750557954849835391121715519, 11.47397939094513156552193260570
