L(s) = 1 | + (−0.707 − 0.707i)3-s + (−2.18 + 0.466i)5-s − 1.00·7-s + 1.00i·9-s + (−1.89 − 1.89i)11-s + (−2.65 − 2.65i)13-s + (1.87 + 1.21i)15-s − 1.73i·17-s + (5.33 − 5.33i)19-s + (0.707 + 0.707i)21-s − 0.160·23-s + (4.56 − 2.04i)25-s + (0.707 − 0.707i)27-s + (−2.70 + 2.70i)29-s + 4.64·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.977 + 0.208i)5-s − 0.378·7-s + 0.333i·9-s + (−0.571 − 0.571i)11-s + (−0.737 − 0.737i)13-s + (0.484 + 0.314i)15-s − 0.421i·17-s + (1.22 − 1.22i)19-s + (0.154 + 0.154i)21-s − 0.0334·23-s + (0.912 − 0.408i)25-s + (0.136 − 0.136i)27-s + (−0.501 + 0.501i)29-s + 0.833·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4096373762\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4096373762\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.18 - 0.466i)T \) |
good | 7 | \( 1 + 1.00T + 7T^{2} \) |
| 11 | \( 1 + (1.89 + 1.89i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.65 + 2.65i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 + (-5.33 + 5.33i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.160T + 23T^{2} \) |
| 29 | \( 1 + (2.70 - 2.70i)T - 29iT^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 + (5.35 - 5.35i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.89iT - 41T^{2} \) |
| 43 | \( 1 + (7.23 - 7.23i)T - 43iT^{2} \) |
| 47 | \( 1 - 4.79iT - 47T^{2} \) |
| 53 | \( 1 + (3.44 - 3.44i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.101 - 0.101i)T + 59iT^{2} \) |
| 61 | \( 1 + (-6.01 + 6.01i)T - 61iT^{2} \) |
| 67 | \( 1 + (-9.04 - 9.04i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.60iT - 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 5.73T + 79T^{2} \) |
| 83 | \( 1 + (-2.04 - 2.04i)T + 83iT^{2} \) |
| 89 | \( 1 - 15.0iT - 89T^{2} \) |
| 97 | \( 1 + 3.84iT - 97T^{2} \) |
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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
−9.471733727099973920007825678716, −8.323556386667797176368166840513, −7.80555156007164314243970686417, −7.05926215130214925775118030997, −6.38840353179855763946334456840, −5.19598118653673097939190625997, −4.74733087267782618973569895195, −3.22402832542570217767818501372, −2.83719132006045557392148933003, −0.941954583576484854313372959895,
0.19958479738870074635027238801, 1.93208824076930946678389142842, 3.37043637096836427528197694672, 4.01983782962316714163736683289, 4.96366560811190199087679439064, 5.61756260725030014278952339172, 6.81534530376046345698815226105, 7.40368482476761386094510836176, 8.194396948249690491768663152378, 9.072818424985707233789544947715