| L(s) = 1 | − 4·2-s + 24·3-s + 16·4-s + 25·5-s − 96·6-s − 172·7-s − 64·8-s + 333·9-s − 100·10-s + 132·11-s + 384·12-s − 946·13-s + 688·14-s + 600·15-s + 256·16-s − 222·17-s − 1.33e3·18-s + 500·19-s + 400·20-s − 4.12e3·21-s − 528·22-s + 3.56e3·23-s − 1.53e3·24-s + 625·25-s + 3.78e3·26-s + 2.16e3·27-s − 2.75e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.53·3-s + 1/2·4-s + 0.447·5-s − 1.08·6-s − 1.32·7-s − 0.353·8-s + 1.37·9-s − 0.316·10-s + 0.328·11-s + 0.769·12-s − 1.55·13-s + 0.938·14-s + 0.688·15-s + 1/4·16-s − 0.186·17-s − 0.968·18-s + 0.317·19-s + 0.223·20-s − 2.04·21-s − 0.232·22-s + 1.40·23-s − 0.544·24-s + 1/5·25-s + 1.09·26-s + 0.570·27-s − 0.663·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.239525664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.239525664\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 8 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 172 T + p^{5} T^{2} \) |
| 11 | \( 1 - 12 p T + p^{5} T^{2} \) |
| 13 | \( 1 + 946 T + p^{5} T^{2} \) |
| 17 | \( 1 + 222 T + p^{5} T^{2} \) |
| 19 | \( 1 - 500 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3564 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2190 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2312 T + p^{5} T^{2} \) |
| 37 | \( 1 + 11242 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1242 T + p^{5} T^{2} \) |
| 43 | \( 1 - 20624 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6588 T + p^{5} T^{2} \) |
| 53 | \( 1 + 21066 T + p^{5} T^{2} \) |
| 59 | \( 1 - 7980 T + p^{5} T^{2} \) |
| 61 | \( 1 - 16622 T + p^{5} T^{2} \) |
| 67 | \( 1 - 1808 T + p^{5} T^{2} \) |
| 71 | \( 1 + 24528 T + p^{5} T^{2} \) |
| 73 | \( 1 - 20474 T + p^{5} T^{2} \) |
| 79 | \( 1 + 46240 T + p^{5} T^{2} \) |
| 83 | \( 1 + 51576 T + p^{5} T^{2} \) |
| 89 | \( 1 + 110310 T + p^{5} T^{2} \) |
| 97 | \( 1 + 78382 T + p^{5} T^{2} \) |
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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
−19.59391109551468014927886506832, −19.12123466791784322961257894419, −17.18326538768877506878988180891, −15.58982682437729295459342374829, −14.22814262916776208556752957778, −12.71552430934689029751480805956, −9.914723701378331839439863103411, −9.025882871241575093930837269361, −7.12322107097916313080871502614, −2.80721942397989604211213493727,
2.80721942397989604211213493727, 7.12322107097916313080871502614, 9.025882871241575093930837269361, 9.914723701378331839439863103411, 12.71552430934689029751480805956, 14.22814262916776208556752957778, 15.58982682437729295459342374829, 17.18326538768877506878988180891, 19.12123466791784322961257894419, 19.59391109551468014927886506832
