| L(s) = 1 | − 32·2-s − 243·3-s + 1.02e3·4-s + 7.77e3·6-s − 3.29e4·7-s − 3.27e4·8-s + 5.90e4·9-s − 7.58e5·11-s − 2.48e5·12-s + 2.48e6·13-s + 1.05e6·14-s + 1.04e6·16-s − 8.29e6·17-s − 1.88e6·18-s − 1.08e7·19-s + 8.00e6·21-s + 2.42e7·22-s − 2.05e7·23-s + 7.96e6·24-s − 7.94e7·26-s − 1.43e7·27-s − 3.37e7·28-s + 2.88e7·29-s + 1.50e8·31-s − 3.35e7·32-s + 1.84e8·33-s + 2.65e8·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.740·7-s − 0.353·8-s + 1/3·9-s − 1.42·11-s − 0.288·12-s + 1.85·13-s + 0.523·14-s + 1/4·16-s − 1.41·17-s − 0.235·18-s − 1.00·19-s + 0.427·21-s + 1.00·22-s − 0.665·23-s + 0.204·24-s − 1.31·26-s − 0.192·27-s − 0.370·28-s + 0.260·29-s + 0.944·31-s − 0.176·32-s + 0.820·33-s + 1.00·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.4422729502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4422729502\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{5} T \) |
| 3 | \( 1 + p^{5} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 32936 T + p^{11} T^{2} \) |
| 11 | \( 1 + 758748 T + p^{11} T^{2} \) |
| 13 | \( 1 - 2482858 T + p^{11} T^{2} \) |
| 17 | \( 1 + 8290386 T + p^{11} T^{2} \) |
| 19 | \( 1 + 10867300 T + p^{11} T^{2} \) |
| 23 | \( 1 + 20539272 T + p^{11} T^{2} \) |
| 29 | \( 1 - 28814550 T + p^{11} T^{2} \) |
| 31 | \( 1 - 150501392 T + p^{11} T^{2} \) |
| 37 | \( 1 - 8645722 p T + p^{11} T^{2} \) |
| 41 | \( 1 + 368008998 T + p^{11} T^{2} \) |
| 43 | \( 1 + 620469572 T + p^{11} T^{2} \) |
| 47 | \( 1 + 2763110256 T + p^{11} T^{2} \) |
| 53 | \( 1 - 268284258 T + p^{11} T^{2} \) |
| 59 | \( 1 - 1672894740 T + p^{11} T^{2} \) |
| 61 | \( 1 + 7787197498 T + p^{11} T^{2} \) |
| 67 | \( 1 + 18706694156 T + p^{11} T^{2} \) |
| 71 | \( 1 + 8346990888 T + p^{11} T^{2} \) |
| 73 | \( 1 + 19641746522 T + p^{11} T^{2} \) |
| 79 | \( 1 + 5873807200 T + p^{11} T^{2} \) |
| 83 | \( 1 + 8492558172 T + p^{11} T^{2} \) |
| 89 | \( 1 - 75527864010 T + p^{11} T^{2} \) |
| 97 | \( 1 - 82356782494 T + p^{11} T^{2} \) |
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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
−10.76005485732786193333903338172, −10.14232091097595963234289042221, −8.827076715795658522069799488524, −8.022536562809858036753374282923, −6.56607966442649288587107092926, −6.04220933144258190984858582892, −4.49869696222175789495463639439, −3.06318094364371104717337454373, −1.76182215492518012092477009204, −0.34795056068503844723200204445,
0.34795056068503844723200204445, 1.76182215492518012092477009204, 3.06318094364371104717337454373, 4.49869696222175789495463639439, 6.04220933144258190984858582892, 6.56607966442649288587107092926, 8.022536562809858036753374282923, 8.827076715795658522069799488524, 10.14232091097595963234289042221, 10.76005485732786193333903338172
