| L(s) = 1 | + (−1 − i)2-s + (−1 + i)3-s + 2i·4-s + 2·6-s + (3 + 3i)7-s + (2 − 2i)8-s + i·9-s + (−2 − 2i)12-s − 6i·14-s − 4·16-s + (1 − i)18-s − 6·21-s + (−1 + i)23-s + 4i·24-s + (−4 − 4i)27-s + (−6 + 6i)28-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.577 + 0.577i)3-s + i·4-s + 0.816·6-s + (1.13 + 1.13i)7-s + (0.707 − 0.707i)8-s + 0.333i·9-s + (−0.577 − 0.577i)12-s − 1.60i·14-s − 16-s + (0.235 − 0.235i)18-s − 1.30·21-s + (−0.208 + 0.208i)23-s + 0.816i·24-s + (−0.769 − 0.769i)27-s + (−1.13 + 1.13i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.632226 + 0.179602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.632226 + 0.179602i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 + i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + (-3 - 3i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (1 - i)T - 23iT^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + (-9 + 9i)T - 43iT^{2} \) |
| 47 | \( 1 + (7 + 7i)T + 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + (-3 - 3i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (11 - 11i)T - 83iT^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−13.87330654053294093103732835354, −12.46699135638758000506690151170, −11.54863093862060596469835855300, −10.94106370616797577259549842392, −9.789717722059306774961706952340, −8.660462326957452758467842797519, −7.68329137688981672666065719211, −5.65376729005721891259866567380, −4.37281928603781804724642807526, −2.24094499277926967581441960442,
1.20653667866600412261009118232, 4.57384591056200679175988775452, 6.01427391829253350775252826613, 7.16925509825480586064625391349, 7.946924863914759800806036868547, 9.330218734313939851695959633623, 10.69883835230387282695339425485, 11.34477273237301703401121060861, 12.77714790246022566882270834816, 14.11858105468047273825572995671
