L(s) = 1 | − 2.88·3-s − 1.00·5-s − 1.34·7-s + 5.30·9-s − 0.428·11-s + 2.65·13-s + 2.88·15-s + 8.02·17-s + 4.96·19-s + 3.87·21-s − 7.95·23-s − 3.99·25-s − 6.64·27-s + 7.89·29-s − 1.02·31-s + 1.23·33-s + 1.34·35-s − 6.45·37-s − 7.63·39-s − 0.415·41-s − 2.10·43-s − 5.31·45-s − 2.82·47-s − 5.18·49-s − 23.1·51-s − 10.7·53-s + 0.429·55-s + ⋯ |
L(s) = 1 | − 1.66·3-s − 0.448·5-s − 0.508·7-s + 1.76·9-s − 0.129·11-s + 0.735·13-s + 0.745·15-s + 1.94·17-s + 1.13·19-s + 0.846·21-s − 1.65·23-s − 0.799·25-s − 1.27·27-s + 1.46·29-s − 0.184·31-s + 0.214·33-s + 0.227·35-s − 1.06·37-s − 1.22·39-s − 0.0649·41-s − 0.321·43-s − 0.792·45-s − 0.412·47-s − 0.741·49-s − 3.23·51-s − 1.47·53-s + 0.0578·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7737009888\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7737009888\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + 2.88T + 3T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 7 | \( 1 + 1.34T + 7T^{2} \) |
| 11 | \( 1 + 0.428T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 - 8.02T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 + 7.95T + 23T^{2} \) |
| 29 | \( 1 - 7.89T + 29T^{2} \) |
| 31 | \( 1 + 1.02T + 31T^{2} \) |
| 37 | \( 1 + 6.45T + 37T^{2} \) |
| 41 | \( 1 + 0.415T + 41T^{2} \) |
| 43 | \( 1 + 2.10T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 2.97T + 59T^{2} \) |
| 61 | \( 1 - 9.70T + 61T^{2} \) |
| 67 | \( 1 + 1.01T + 67T^{2} \) |
| 71 | \( 1 - 6.32T + 71T^{2} \) |
| 73 | \( 1 - 4.36T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 9.87T + 83T^{2} \) |
| 89 | \( 1 + 6.59T + 89T^{2} \) |
| 97 | \( 1 + 2.55T + 97T^{2} \) |
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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
−8.100434650692009422643163188762, −7.76088902827070358773140631907, −6.71306721181095733890857071196, −6.19147848519694036707605009241, −5.50917437317870558591042462814, −4.95203914522768902794655943913, −3.85404786643648836425188781290, −3.25669547262563174304118118550, −1.55561111834002063826894763743, −0.57949135531406433370112201387,
0.57949135531406433370112201387, 1.55561111834002063826894763743, 3.25669547262563174304118118550, 3.85404786643648836425188781290, 4.95203914522768902794655943913, 5.50917437317870558591042462814, 6.19147848519694036707605009241, 6.71306721181095733890857071196, 7.76088902827070358773140631907, 8.100434650692009422643163188762