| L(s) = 1 | − 4i·2-s + 9i·3-s − 16·4-s + 36·6-s − 49i·7-s + 64i·8-s + 162·9-s − 187·11-s − 144i·12-s − 627i·13-s − 196·14-s + 256·16-s + 1.81e3i·17-s − 648i·18-s − 258·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.377i·7-s + 0.353i·8-s + 0.666·9-s − 0.465·11-s − 0.288i·12-s − 1.02i·13-s − 0.267·14-s + 0.250·16-s + 1.52i·17-s − 0.471i·18-s − 0.163·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.420245254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420245254\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49iT \) |
| good | 3 | \( 1 - 9iT - 243T^{2} \) |
| 11 | \( 1 + 187T + 1.61e5T^{2} \) |
| 13 | \( 1 + 627iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.81e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 258T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.97e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.57e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 6.67e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.17e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.20e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.61e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.93e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.87e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.48e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.37e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.04e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 4.37e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.72e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 4.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.72e4iT - 8.58e9T^{2} \) |
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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
−10.47906732428098559445644187128, −9.800798676645035956057119740927, −8.586251534171562978272507600477, −7.77568503572756408848118064551, −6.39280228010322361253858911285, −5.11567945929523758931851025246, −4.17578634295150094482908847181, −3.23190278568680553878919085783, −1.79968063732603726623319593322, −0.40158329257998323542529266675,
1.14858579790761004906671429919, 2.53655569775089899785250632122, 4.16030512393228373237840730476, 5.20304712749982916182773508495, 6.30866954542946591471542613957, 7.24036024944951791779626823826, 7.80849775331908746972892981901, 9.145382413939012334237956031089, 9.665237736431242843906902789257, 11.05699028783161484141450211078
