| L(s) = 1 | + 9·3-s + 25·5-s − 128·7-s + 81·9-s + 308·11-s − 1.05e3·13-s + 225·15-s + 1.58e3·17-s − 2.30e3·19-s − 1.15e3·21-s − 2.65e3·23-s + 625·25-s + 729·27-s + 1.19e3·29-s − 9.52e3·31-s + 2.77e3·33-s − 3.20e3·35-s + 4.47e3·37-s − 9.52e3·39-s − 6.19e3·41-s + 6.33e3·43-s + 2.02e3·45-s − 1.49e4·47-s − 423·49-s + 1.42e4·51-s + 3.83e4·53-s + 7.70e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.987·7-s + 1/3·9-s + 0.767·11-s − 1.73·13-s + 0.258·15-s + 1.33·17-s − 1.46·19-s − 0.570·21-s − 1.04·23-s + 1/5·25-s + 0.192·27-s + 0.264·29-s − 1.77·31-s + 0.443·33-s − 0.441·35-s + 0.536·37-s − 1.00·39-s − 0.575·41-s + 0.522·43-s + 0.149·45-s − 0.985·47-s − 0.0251·49-s + 0.768·51-s + 1.87·53-s + 0.343·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 + 128 T + p^{5} T^{2} \) |
| 11 | \( 1 - 28 p T + p^{5} T^{2} \) |
| 13 | \( 1 + 1058 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1586 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2308 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2656 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1198 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9520 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4470 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6198 T + p^{5} T^{2} \) |
| 43 | \( 1 - 6332 T + p^{5} T^{2} \) |
| 47 | \( 1 + 14920 T + p^{5} T^{2} \) |
| 53 | \( 1 - 38310 T + p^{5} T^{2} \) |
| 59 | \( 1 + 196 p T + p^{5} T^{2} \) |
| 61 | \( 1 + 48338 T + p^{5} T^{2} \) |
| 67 | \( 1 + 56972 T + p^{5} T^{2} \) |
| 71 | \( 1 + 44856 T + p^{5} T^{2} \) |
| 73 | \( 1 + 19446 T + p^{5} T^{2} \) |
| 79 | \( 1 - 77328 T + p^{5} T^{2} \) |
| 83 | \( 1 + 40364 T + p^{5} T^{2} \) |
| 89 | \( 1 - 35706 T + p^{5} T^{2} \) |
| 97 | \( 1 + 97022 T + p^{5} T^{2} \) |
| 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
−10.43919944264366929671717721247, −9.749334674666883777118237462515, −9.057313986570085744618334608117, −7.76013281051998384128075926963, −6.79022568638182470435477701747, −5.72959178139775692632875867481, −4.26208040865700683696526503034, −3.04989953045417799470428930496, −1.86436847897045882467438207660, 0,
1.86436847897045882467438207660, 3.04989953045417799470428930496, 4.26208040865700683696526503034, 5.72959178139775692632875867481, 6.79022568638182470435477701747, 7.76013281051998384128075926963, 9.057313986570085744618334608117, 9.749334674666883777118237462515, 10.43919944264366929671717721247
