| L(s) = 1 | − 2-s − 2.31·3-s + 4-s + 5-s + 2.31·6-s + 4.25·7-s − 8-s + 2.37·9-s − 10-s + 11-s − 2.31·12-s + 2.14·13-s − 4.25·14-s − 2.31·15-s + 16-s + 0.399·17-s − 2.37·18-s − 0.398·19-s + 20-s − 9.85·21-s − 22-s − 1.33·23-s + 2.31·24-s + 25-s − 2.14·26-s + 1.45·27-s + 4.25·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.33·3-s + 0.5·4-s + 0.447·5-s + 0.946·6-s + 1.60·7-s − 0.353·8-s + 0.790·9-s − 0.316·10-s + 0.301·11-s − 0.669·12-s + 0.595·13-s − 1.13·14-s − 0.598·15-s + 0.250·16-s + 0.0969·17-s − 0.559·18-s − 0.0914·19-s + 0.223·20-s − 2.15·21-s − 0.213·22-s − 0.277·23-s + 0.473·24-s + 0.200·25-s − 0.421·26-s + 0.279·27-s + 0.803·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.249676341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.249676341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 2.31T + 3T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 - 0.399T + 17T^{2} \) |
| 19 | \( 1 + 0.398T + 19T^{2} \) |
| 23 | \( 1 + 1.33T + 23T^{2} \) |
| 29 | \( 1 + 4.75T + 29T^{2} \) |
| 31 | \( 1 - 7.39T + 31T^{2} \) |
| 37 | \( 1 - 0.118T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 47 | \( 1 + 3.10T + 47T^{2} \) |
| 53 | \( 1 - 6.55T + 53T^{2} \) |
| 59 | \( 1 + 3.06T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 5.14T + 73T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 5.68T + 97T^{2} \) |
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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
−8.313126129063096719554358835432, −7.60741525981249303107328923608, −6.81236843236228472875390703003, −6.02521262109244354278884204777, −5.55859856583602238251007711278, −4.77954174078229624376257404093, −4.00918509716241890541387902632, −2.51003255113544496841817516575, −1.52640497433050082709337696877, −0.811668808656325173656208869953,
0.811668808656325173656208869953, 1.52640497433050082709337696877, 2.51003255113544496841817516575, 4.00918509716241890541387902632, 4.77954174078229624376257404093, 5.55859856583602238251007711278, 6.02521262109244354278884204777, 6.81236843236228472875390703003, 7.60741525981249303107328923608, 8.313126129063096719554358835432
