| L(s) = 1 | + 2.27·2-s − 3·3-s − 2.83·4-s + 4.25·5-s − 6.81·6-s − 24.6·8-s + 9·9-s + 9.66·10-s + 8.37·11-s + 8.51·12-s + 13·13-s − 12.7·15-s − 33.2·16-s − 46.7·17-s + 20.4·18-s + 34.3·19-s − 12.0·20-s + 19.0·22-s + 185.·23-s + 73.8·24-s − 106.·25-s + 29.5·26-s − 27·27-s + 34.1·29-s − 28.9·30-s − 294.·31-s + 121.·32-s + ⋯ |
| L(s) = 1 | + 0.803·2-s − 0.577·3-s − 0.354·4-s + 0.380·5-s − 0.463·6-s − 1.08·8-s + 0.333·9-s + 0.305·10-s + 0.229·11-s + 0.204·12-s + 0.277·13-s − 0.219·15-s − 0.519·16-s − 0.666·17-s + 0.267·18-s + 0.414·19-s − 0.135·20-s + 0.184·22-s + 1.67·23-s + 0.628·24-s − 0.855·25-s + 0.222·26-s − 0.192·27-s + 0.218·29-s − 0.176·30-s − 1.70·31-s + 0.671·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 - 2.27T + 8T^{2} \) |
| 5 | \( 1 - 4.25T + 125T^{2} \) |
| 11 | \( 1 - 8.37T + 1.33e3T^{2} \) |
| 17 | \( 1 + 46.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 34.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 294.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 31.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 233.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 39.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 430.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 661.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 727.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 489.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 54.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 232.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 891.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 628.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 985.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 830.T + 9.12e5T^{2} \) |
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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
−8.678782948581720787472908096047, −7.43982354468236319976595752462, −6.65940782638575664207063150405, −5.78478434133614866300918959244, −5.30379935584567737347456556752, −4.40690412658401879525520591567, −3.65790133178992521282100138500, −2.57975394183153751720603640637, −1.22583477028210366413334370214, 0,
1.22583477028210366413334370214, 2.57975394183153751720603640637, 3.65790133178992521282100138500, 4.40690412658401879525520591567, 5.30379935584567737347456556752, 5.78478434133614866300918959244, 6.65940782638575664207063150405, 7.43982354468236319976595752462, 8.678782948581720787472908096047
