| L(s) = 1 | + 0.190·2-s − 1.43·3-s − 7.96·4-s − 16.3·5-s − 0.274·6-s − 6.23·7-s − 3.04·8-s − 24.9·9-s − 3.12·10-s − 29.4·11-s + 11.4·12-s + 22.2·13-s − 1.19·14-s + 23.5·15-s + 63.1·16-s − 60.3·17-s − 4.75·18-s + 10.0·19-s + 130.·20-s + 8.97·21-s − 5.62·22-s − 40.6·23-s + 4.38·24-s + 142.·25-s + 4.24·26-s + 74.6·27-s + 49.6·28-s + ⋯ |
| L(s) = 1 | + 0.0674·2-s − 0.276·3-s − 0.995·4-s − 1.46·5-s − 0.0186·6-s − 0.336·7-s − 0.134·8-s − 0.923·9-s − 0.0986·10-s − 0.807·11-s + 0.275·12-s + 0.474·13-s − 0.0227·14-s + 0.404·15-s + 0.986·16-s − 0.861·17-s − 0.0623·18-s + 0.121·19-s + 1.45·20-s + 0.0932·21-s − 0.0544·22-s − 0.368·23-s + 0.0372·24-s + 1.13·25-s + 0.0320·26-s + 0.532·27-s + 0.335·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 0.190T + 8T^{2} \) |
| 3 | \( 1 + 1.43T + 27T^{2} \) |
| 5 | \( 1 + 16.3T + 125T^{2} \) |
| 7 | \( 1 + 6.23T + 343T^{2} \) |
| 11 | \( 1 + 29.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 60.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 40.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 195.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 240.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 172.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 583.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 370.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 714.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 707.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 377.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 47.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 691.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 288.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 833.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 687.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−8.428285405061512584771105297397, −7.999916736668971755089080968372, −6.94469978140213737937364261271, −6.00214640245264606722659799489, −5.07067064562817986197095148958, −4.37772130708394874367527521624, −3.55265415010338364866353513807, −2.72197560016556489821955231221, −0.73185095126233999928580591029, 0,
0.73185095126233999928580591029, 2.72197560016556489821955231221, 3.55265415010338364866353513807, 4.37772130708394874367527521624, 5.07067064562817986197095148958, 6.00214640245264606722659799489, 6.94469978140213737937364261271, 7.999916736668971755089080968372, 8.428285405061512584771105297397
