| L(s) = 1 | + 2.18e3·3-s − 3.14e5·5-s − 2.02e6·7-s + 4.78e6·9-s − 1.10e8·11-s + 5.60e7·13-s − 6.87e8·15-s − 1.93e9·17-s − 2.16e9·19-s − 4.42e9·21-s − 6.22e9·23-s + 6.83e10·25-s + 1.04e10·27-s + 6.47e10·29-s + 2.02e10·31-s − 2.41e11·33-s + 6.36e11·35-s + 4.88e11·37-s + 1.22e11·39-s − 7.72e11·41-s − 1.30e12·43-s − 1.50e12·45-s − 3.35e12·47-s − 6.46e11·49-s − 4.22e12·51-s + 9.38e12·53-s + 3.46e13·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.80·5-s − 0.929·7-s + 1/3·9-s − 1.70·11-s + 0.247·13-s − 1.03·15-s − 1.14·17-s − 0.555·19-s − 0.536·21-s − 0.381·23-s + 2.24·25-s + 0.192·27-s + 0.696·29-s + 0.132·31-s − 0.984·33-s + 1.67·35-s + 0.846·37-s + 0.143·39-s − 0.619·41-s − 0.733·43-s − 0.600·45-s − 0.965·47-s − 0.136·49-s − 0.658·51-s + 1.09·53-s + 3.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.5579053845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5579053845\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{7} T \) |
| good | 5 | \( 1 + 62898 p T + p^{15} T^{2} \) |
| 7 | \( 1 + 2025056 T + p^{15} T^{2} \) |
| 11 | \( 1 + 110255052 T + p^{15} T^{2} \) |
| 13 | \( 1 - 4311374 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 1930104414 T + p^{15} T^{2} \) |
| 19 | \( 1 + 2163188180 T + p^{15} T^{2} \) |
| 23 | \( 1 + 6228974472 T + p^{15} T^{2} \) |
| 29 | \( 1 - 64743719070 T + p^{15} T^{2} \) |
| 31 | \( 1 - 20237611048 T + p^{15} T^{2} \) |
| 37 | \( 1 - 488967594446 T + p^{15} T^{2} \) |
| 41 | \( 1 + 772359114198 T + p^{15} T^{2} \) |
| 43 | \( 1 + 1306766329292 T + p^{15} T^{2} \) |
| 47 | \( 1 + 3351821491776 T + p^{15} T^{2} \) |
| 53 | \( 1 - 9387813393702 T + p^{15} T^{2} \) |
| 59 | \( 1 + 28930359275340 T + p^{15} T^{2} \) |
| 61 | \( 1 - 42393077399702 T + p^{15} T^{2} \) |
| 67 | \( 1 - 52247243064364 T + p^{15} T^{2} \) |
| 71 | \( 1 - 27194529024648 T + p^{15} T^{2} \) |
| 73 | \( 1 + 91604195687878 T + p^{15} T^{2} \) |
| 79 | \( 1 + 62882111078120 T + p^{15} T^{2} \) |
| 83 | \( 1 - 223567315949868 T + p^{15} T^{2} \) |
| 89 | \( 1 - 554198786115210 T + p^{15} T^{2} \) |
| 97 | \( 1 + 14318252338942 p T + p^{15} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.61080966630491615173497034947, −11.31013995331739065951957147310, −10.20257058131096222521287503989, −8.591654794432761078572028569193, −7.85151947230228236513215927094, −6.68767939195099554715335075460, −4.70031470350661197985389502440, −3.57393262518244915560134519434, −2.57103729155596431768826548153, −0.35653131254624719802162319947,
0.35653131254624719802162319947, 2.57103729155596431768826548153, 3.57393262518244915560134519434, 4.70031470350661197985389502440, 6.68767939195099554715335075460, 7.85151947230228236513215927094, 8.591654794432761078572028569193, 10.20257058131096222521287503989, 11.31013995331739065951957147310, 12.61080966630491615173497034947
