L(s) = 1 | − 2-s + 2.36·3-s + 4-s − 0.759·5-s − 2.36·6-s − 2.98·7-s − 8-s + 2.60·9-s + 0.759·10-s + 11-s + 2.36·12-s + 5.18·13-s + 2.98·14-s − 1.79·15-s + 16-s + 5.91·17-s − 2.60·18-s − 6.38·19-s − 0.759·20-s − 7.07·21-s − 22-s + 0.346·23-s − 2.36·24-s − 4.42·25-s − 5.18·26-s − 0.937·27-s − 2.98·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.36·3-s + 0.5·4-s − 0.339·5-s − 0.966·6-s − 1.13·7-s − 0.353·8-s + 0.867·9-s + 0.240·10-s + 0.301·11-s + 0.683·12-s + 1.43·13-s + 0.799·14-s − 0.464·15-s + 0.250·16-s + 1.43·17-s − 0.613·18-s − 1.46·19-s − 0.169·20-s − 1.54·21-s − 0.213·22-s + 0.0722·23-s − 0.483·24-s − 0.884·25-s − 1.01·26-s − 0.180·27-s − 0.565·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.955956963\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.955956963\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
good | 3 | \( 1 - 2.36T + 3T^{2} \) |
| 5 | \( 1 + 0.759T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 - 5.91T + 17T^{2} \) |
| 19 | \( 1 + 6.38T + 19T^{2} \) |
| 23 | \( 1 - 0.346T + 23T^{2} \) |
| 29 | \( 1 + 0.999T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 - 9.58T + 37T^{2} \) |
| 41 | \( 1 - 2.84T + 41T^{2} \) |
| 43 | \( 1 - 5.35T + 43T^{2} \) |
| 47 | \( 1 - 8.37T + 47T^{2} \) |
| 53 | \( 1 - 5.58T + 53T^{2} \) |
| 59 | \( 1 + 0.419T + 59T^{2} \) |
| 61 | \( 1 + 5.87T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 7.25T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 7.40T + 79T^{2} \) |
| 83 | \( 1 - 2.07T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 6.10T + 97T^{2} \) |
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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
−8.346796700859677773440182937697, −7.914606094803990572532731551842, −7.20338756156285825761206893904, −6.23530367851039696876397562334, −5.83130876419354299825700380509, −4.07211479101216596435588752767, −3.67673052895441980173946338370, −2.90802332582345551477336879899, −2.03713707770588902964022449448, −0.820434378478122163520623753388,
0.820434378478122163520623753388, 2.03713707770588902964022449448, 2.90802332582345551477336879899, 3.67673052895441980173946338370, 4.07211479101216596435588752767, 5.83130876419354299825700380509, 6.23530367851039696876397562334, 7.20338756156285825761206893904, 7.914606094803990572532731551842, 8.346796700859677773440182937697