| L(s) = 1 | − 27·3-s + 125·5-s + 1.40e3·7-s + 729·9-s + 4.04e3·11-s − 5.89e3·13-s − 3.37e3·15-s + 3.10e4·17-s + 4.03e4·19-s − 3.80e4·21-s + 7.89e4·23-s + 1.56e4·25-s − 1.96e4·27-s − 1.57e5·29-s − 1.14e5·31-s − 1.09e5·33-s + 1.76e5·35-s − 4.71e5·37-s + 1.59e5·39-s − 4.04e5·41-s + 2.53e5·43-s + 9.11e4·45-s − 4.37e5·47-s + 1.15e6·49-s − 8.37e5·51-s + 3.34e5·53-s + 5.05e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.55·7-s + 1/3·9-s + 0.916·11-s − 0.743·13-s − 0.258·15-s + 1.53·17-s + 1.34·19-s − 0.895·21-s + 1.35·23-s + 1/5·25-s − 0.192·27-s − 1.19·29-s − 0.692·31-s − 0.528·33-s + 0.693·35-s − 1.53·37-s + 0.429·39-s − 0.916·41-s + 0.486·43-s + 0.149·45-s − 0.614·47-s + 1.40·49-s − 0.883·51-s + 0.309·53-s + 0.409·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.781921563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.781921563\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| 5 | \( 1 - p^{3} T \) |
| good | 7 | \( 1 - 1408 T + p^{7} T^{2} \) |
| 11 | \( 1 - 4044 T + p^{7} T^{2} \) |
| 13 | \( 1 + 5890 T + p^{7} T^{2} \) |
| 17 | \( 1 - 31002 T + p^{7} T^{2} \) |
| 19 | \( 1 - 40300 T + p^{7} T^{2} \) |
| 23 | \( 1 - 78912 T + p^{7} T^{2} \) |
| 29 | \( 1 + 157194 T + p^{7} T^{2} \) |
| 31 | \( 1 + 3704 p T + p^{7} T^{2} \) |
| 37 | \( 1 + 471994 T + p^{7} T^{2} \) |
| 41 | \( 1 + 404310 T + p^{7} T^{2} \) |
| 43 | \( 1 - 253852 T + p^{7} T^{2} \) |
| 47 | \( 1 + 437688 T + p^{7} T^{2} \) |
| 53 | \( 1 - 334926 T + p^{7} T^{2} \) |
| 59 | \( 1 + 562596 T + p^{7} T^{2} \) |
| 61 | \( 1 - 3246662 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3895148 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2345160 T + p^{7} T^{2} \) |
| 73 | \( 1 - 5726954 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5222008 T + p^{7} T^{2} \) |
| 83 | \( 1 - 2928132 T + p^{7} T^{2} \) |
| 89 | \( 1 + 3160230 T + p^{7} T^{2} \) |
| 97 | \( 1 + 1898686 T + p^{7} T^{2} \) |
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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
−11.04348464686370767699141247176, −9.950539247164196122483685681750, −9.044649738406282416908099406162, −7.75736531757074696164805966226, −6.98606894726246529228652518213, −5.41703456680094371975914025638, −5.06717257165342080949224187379, −3.52273784841209624245421040435, −1.78096634798540549634832031793, −0.976652657342193778156975566426,
0.976652657342193778156975566426, 1.78096634798540549634832031793, 3.52273784841209624245421040435, 5.06717257165342080949224187379, 5.41703456680094371975914025638, 6.98606894726246529228652518213, 7.75736531757074696164805966226, 9.044649738406282416908099406162, 9.950539247164196122483685681750, 11.04348464686370767699141247176
