| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s + 4.82·7-s − 3·8-s + 9-s + 10-s − 4·11-s − 12-s + 5.41·13-s + 4.82·14-s + 15-s − 16-s + 1.41·17-s + 18-s + 2.24·19-s − 20-s + 4.82·21-s − 4·22-s + 1.17·23-s − 3·24-s + 25-s + 5.41·26-s + 27-s − 4.82·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.82·7-s − 1.06·8-s + 0.333·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 1.50·13-s + 1.29·14-s + 0.258·15-s − 0.250·16-s + 0.342·17-s + 0.235·18-s + 0.514·19-s − 0.223·20-s + 1.05·21-s − 0.852·22-s + 0.244·23-s − 0.612·24-s + 0.200·25-s + 1.06·26-s + 0.192·27-s − 0.912·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.269220373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.269220373\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 5.41T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + 0.242T + 37T^{2} \) |
| 41 | \( 1 - 0.343T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 9.89T + 53T^{2} \) |
| 59 | \( 1 + 7.41T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 4.48T + 67T^{2} \) |
| 71 | \( 1 + 7.89T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 7.41T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 7.31T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.084434442654227120434756815495, −7.67894241326368992001723520895, −6.48663388732873319491872932914, −5.64410347443394617679032525300, −5.09739035044118065689555487808, −4.59281163531757707138818800330, −3.68250924307391899271165733990, −2.93988152811205336492728684612, −1.93662292882704021352176023904, −1.02977341255322462151799157154,
1.02977341255322462151799157154, 1.93662292882704021352176023904, 2.93988152811205336492728684612, 3.68250924307391899271165733990, 4.59281163531757707138818800330, 5.09739035044118065689555487808, 5.64410347443394617679032525300, 6.48663388732873319491872932914, 7.67894241326368992001723520895, 8.084434442654227120434756815495
