| L(s) = 1 | + 2.18e3·3-s + 4.57e4·5-s − 1.21e6·7-s + 4.78e6·9-s + 2.68e7·11-s − 1.62e8·13-s + 9.99e7·15-s − 7.43e8·17-s + 4.00e9·19-s − 2.66e9·21-s + 3.00e10·23-s − 2.84e10·25-s + 1.04e10·27-s + 1.90e10·29-s + 4.62e9·31-s + 5.88e10·33-s − 5.56e10·35-s + 6.49e11·37-s − 3.55e11·39-s + 7.90e11·41-s − 1.38e12·43-s + 2.18e11·45-s + 3.93e12·47-s − 3.26e12·49-s − 1.62e12·51-s − 1.34e13·53-s + 1.22e12·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.261·5-s − 0.558·7-s + 1/3·9-s + 0.416·11-s − 0.718·13-s + 0.151·15-s − 0.439·17-s + 1.02·19-s − 0.322·21-s + 1.84·23-s − 0.931·25-s + 0.192·27-s + 0.204·29-s + 0.0301·31-s + 0.240·33-s − 0.146·35-s + 1.12·37-s − 0.414·39-s + 0.633·41-s − 0.779·43-s + 0.0872·45-s + 1.13·47-s − 0.687·49-s − 0.253·51-s − 1.57·53-s + 0.108·55-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
| \(L(8)\) |
\(\approx\) |
\(2.695880337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.695880337\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 - 45702 T + p^{15} T^{2} \) |
| 7 | \( 1 + 173984 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 2445084 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 12506290 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 743272542 T + p^{15} T^{2} \) |
| 19 | \( 1 - 4003014700 T + p^{15} T^{2} \) |
| 23 | \( 1 - 30097540728 T + p^{15} T^{2} \) |
| 29 | \( 1 - 19021888926 T + p^{15} T^{2} \) |
| 31 | \( 1 - 4621552936 T + p^{15} T^{2} \) |
| 37 | \( 1 - 649297928654 T + p^{15} T^{2} \) |
| 41 | \( 1 - 790230862890 T + p^{15} T^{2} \) |
| 43 | \( 1 + 1388728387532 T + p^{15} T^{2} \) |
| 47 | \( 1 - 3933841180608 T + p^{15} T^{2} \) |
| 53 | \( 1 + 13472208095706 T + p^{15} T^{2} \) |
| 59 | \( 1 - 24672598493364 T + p^{15} T^{2} \) |
| 61 | \( 1 - 23630686395542 T + p^{15} T^{2} \) |
| 67 | \( 1 + 32385083278292 T + p^{15} T^{2} \) |
| 71 | \( 1 - 74451150070920 T + p^{15} T^{2} \) |
| 73 | \( 1 - 176524276453946 T + p^{15} T^{2} \) |
| 79 | \( 1 - 137959485182488 T + p^{15} T^{2} \) |
| 83 | \( 1 - 458794939458348 T + p^{15} T^{2} \) |
| 89 | \( 1 + 32239404369270 T + p^{15} T^{2} \) |
| 97 | \( 1 - 478308097627586 T + p^{15} T^{2} \) |
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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
−12.64707364263657028764742348208, −11.32114728445668206404176554929, −9.829712227292944939931671248901, −9.125828557433387514400927989268, −7.64705774038209837024674581230, −6.51687917706388523186185504167, −4.97655921445085556843941156545, −3.48611856803460441262905533427, −2.34946663733829211555159274495, −0.849168262204305172184181129840,
0.849168262204305172184181129840, 2.34946663733829211555159274495, 3.48611856803460441262905533427, 4.97655921445085556843941156545, 6.51687917706388523186185504167, 7.64705774038209837024674581230, 9.125828557433387514400927989268, 9.829712227292944939931671248901, 11.32114728445668206404176554929, 12.64707364263657028764742348208
