| L(s) = 1 | − 2.09·3-s − 0.681·5-s + 4.42·7-s + 1.40·9-s − 3.45·11-s + 0.452·13-s + 1.42·15-s − 4.79·17-s + 2.70·19-s − 9.28·21-s − 1.65·23-s − 4.53·25-s + 3.35·27-s − 5.71·29-s − 1.44·31-s + 7.24·33-s − 3.01·35-s − 5.71·37-s − 0.948·39-s + 3.22·41-s − 3.25·43-s − 0.954·45-s + 12.7·47-s + 12.5·49-s + 10.0·51-s − 6.28·53-s + 2.35·55-s + ⋯ |
| L(s) = 1 | − 1.21·3-s − 0.304·5-s + 1.67·7-s + 0.467·9-s − 1.04·11-s + 0.125·13-s + 0.369·15-s − 1.16·17-s + 0.621·19-s − 2.02·21-s − 0.345·23-s − 0.907·25-s + 0.645·27-s − 1.06·29-s − 0.260·31-s + 1.26·33-s − 0.509·35-s − 0.939·37-s − 0.151·39-s + 0.504·41-s − 0.495·43-s − 0.142·45-s + 1.85·47-s + 1.79·49-s + 1.40·51-s − 0.863·53-s + 0.317·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9172294230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9172294230\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.09T + 3T^{2} \) |
| 5 | \( 1 + 0.681T + 5T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 13 | \( 1 - 0.452T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 + 1.65T + 23T^{2} \) |
| 29 | \( 1 + 5.71T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 + 5.71T + 37T^{2} \) |
| 41 | \( 1 - 3.22T + 41T^{2} \) |
| 43 | \( 1 + 3.25T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 6.28T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 3.81T + 67T^{2} \) |
| 71 | \( 1 - 5.32T + 71T^{2} \) |
| 73 | \( 1 - 5.48T + 73T^{2} \) |
| 79 | \( 1 - 7.64T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 6.80T + 89T^{2} \) |
| 97 | \( 1 + 8.44T + 97T^{2} \) |
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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
−7.73750341850285627362337917560, −7.25329544382678073949717355606, −6.32723153825070155372225373685, −5.48555825444727456177741178472, −5.21982118437165678988764785077, −4.50911503249153792369150538847, −3.76980027711310866158384743312, −2.43792229523950724440122971848, −1.69392958117307474653664647994, −0.50321858005914640238565118861,
0.50321858005914640238565118861, 1.69392958117307474653664647994, 2.43792229523950724440122971848, 3.76980027711310866158384743312, 4.50911503249153792369150538847, 5.21982118437165678988764785077, 5.48555825444727456177741178472, 6.32723153825070155372225373685, 7.25329544382678073949717355606, 7.73750341850285627362337917560
