L(s) = 1 | + (−0.707 + 0.707i)2-s + (−1.70 + 0.292i)3-s − 1.00i·4-s + 2.23i·5-s + (0.999 − 1.41i)6-s + (0.707 + 0.707i)8-s + (2.82 − i)9-s + (−1.58 − 1.58i)10-s − 2.82i·11-s + (0.292 + 1.70i)12-s + (4.16 − 4.16i)13-s + (−0.654 − 3.81i)15-s − 1.00·16-s + (−3.65 + 3.65i)17-s + (−1.29 + 2.70i)18-s − 5.16i·19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.985 + 0.169i)3-s − 0.500i·4-s + 0.999i·5-s + (0.408 − 0.577i)6-s + (0.250 + 0.250i)8-s + (0.942 − 0.333i)9-s + (−0.500 − 0.500i)10-s − 0.852i·11-s + (0.0845 + 0.492i)12-s + (1.15 − 1.15i)13-s + (−0.169 − 0.985i)15-s − 0.250·16-s + (−0.885 + 0.885i)17-s + (−0.304 + 0.638i)18-s − 1.18i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.793138 + 0.300122i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.793138 + 0.300122i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (1.70 - 0.292i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (-4.16 + 4.16i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.65 - 3.65i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.16iT - 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + (-5.16 - 5.16i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.11iT - 41T^{2} \) |
| 43 | \( 1 + (2.83 - 2.83i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.71 + 8.71i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.70 - 6.70i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (-5.16 - 5.16i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.05iT - 71T^{2} \) |
| 73 | \( 1 + (7.32 - 7.32i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.16iT - 79T^{2} \) |
| 83 | \( 1 + (-5.88 - 5.88i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.458T + 89T^{2} \) |
| 97 | \( 1 + (-1.16 - 1.16i)T + 97iT^{2} \) |
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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
−10.58021227125029212408988893705, −9.995711589021767324216728304218, −8.706217301822226190150304070250, −7.982985938250441406383723191320, −6.69698863886307336456486957009, −6.34778465194740519791098965937, −5.48438889915690058936024480027, −4.21148002782277551353241201141, −2.88662618544675750812313032048, −0.872111323977647044875938444791,
0.958778701376155870771859703289, 2.07113355285060370684453634066, 4.13733923143049356280263859201, 4.65728336252795099705528580465, 5.95184332517292251467760929807, 6.82065013906330895846004458035, 7.84914418518137451769623933038, 8.832199136010943656062649961593, 9.552921900557789535247814905568, 10.40659444678048520085799293491