| L(s) = 1 | + (−5.42 − 5.42i)2-s + (34.4 − 34.4i)3-s − 5.14i·4-s − 373.·6-s + (−361. − 361. i)7-s + (−375. + 375. i)8-s − 1.64e3i·9-s + 1.25e3·11-s + (−177. − 177. i)12-s + (−2.74e3 + 2.74e3i)13-s + 3.92e3i·14-s + 3.74e3·16-s + (−206. − 206. i)17-s + (−8.91e3 + 8.91e3i)18-s + 3.84e3i·19-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.678i)2-s + (1.27 − 1.27i)3-s − 0.0803i·4-s − 1.73·6-s + (−1.05 − 1.05i)7-s + (−0.732 + 0.732i)8-s − 2.25i·9-s + 0.942·11-s + (−0.102 − 0.102i)12-s + (−1.24 + 1.24i)13-s + 1.43i·14-s + 0.913·16-s + (−0.0419 − 0.0419i)17-s + (−1.52 + 1.52i)18-s + 0.560i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.9073962107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9073962107\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (5.42 + 5.42i)T + 64iT^{2} \) |
| 3 | \( 1 + (-34.4 + 34.4i)T - 729iT^{2} \) |
| 7 | \( 1 + (361. + 361. i)T + 1.17e5iT^{2} \) |
| 11 | \( 1 - 1.25e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (2.74e3 - 2.74e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + (206. + 206. i)T + 2.41e7iT^{2} \) |
| 19 | \( 1 - 3.84e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (-2.21e3 + 2.21e3i)T - 1.48e8iT^{2} \) |
| 29 | \( 1 + 3.69e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 1.93e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + (3.02e4 + 3.02e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + 6.42e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + (1.15e4 - 1.15e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + (6.19e4 + 6.19e4i)T + 1.07e10iT^{2} \) |
| 53 | \( 1 + (-4.27e4 + 4.27e4i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 - 1.29e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.89e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + (-1.07e5 - 1.07e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 3.84e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (1.81e5 - 1.81e5i)T - 1.51e11iT^{2} \) |
| 79 | \( 1 - 1.33e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (9.16e4 - 9.16e4i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 - 3.83e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (6.09e5 + 6.09e5i)T + 8.32e11iT^{2} \) |
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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
−11.67627813508330000262353784116, −9.947161361404097408659686838889, −9.475934478843389619811214895976, −8.414411134001866135013972191101, −7.13600019188158139212503445949, −6.47492946237151919068233901256, −3.85862132790070644965952211906, −2.53696024497938528803817390488, −1.48797257805861142281469563548, −0.29760309249689258240142852562,
2.77986466116700463552724460333, 3.44235445244178445407679335574, 5.12123258075585533430692822275, 6.74379375994128237493143956857, 8.044710744739569344639410966172, 8.952708715290230580841978626117, 9.465235470095911153858077409688, 10.27274356033767150202377240692, 12.11918426873969670798065953867, 13.07480184102529935848781916734
