| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 4.80i·7-s − i·8-s − 9-s − 0.882·11-s + i·12-s + 2.82i·13-s + 4.80·14-s + 16-s + 1.65i·17-s − i·18-s − 5.37·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.81i·7-s − 0.353i·8-s − 0.333·9-s − 0.266·11-s + 0.288i·12-s + 0.783i·13-s + 1.28·14-s + 0.250·16-s + 0.402i·17-s − 0.235i·18-s − 1.23·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3189960434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3189960434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.80iT - 7T^{2} \) |
| 11 | \( 1 + 0.882T + 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 1.65iT - 17T^{2} \) |
| 19 | \( 1 + 5.37T + 19T^{2} \) |
| 23 | \( 1 + 5.85iT - 23T^{2} \) |
| 29 | \( 1 + 3.57T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 1.74iT - 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 + 2.27iT - 43T^{2} \) |
| 47 | \( 1 + 8.72iT - 47T^{2} \) |
| 53 | \( 1 + 3.54iT - 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 0.0862T + 61T^{2} \) |
| 67 | \( 1 - 11.9iT - 67T^{2} \) |
| 71 | \( 1 + 3.50T + 71T^{2} \) |
| 73 | \( 1 + 7.22iT - 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 13.2iT - 83T^{2} \) |
| 89 | \( 1 + 18.6T + 89T^{2} \) |
| 97 | \( 1 - 18.2iT - 97T^{2} \) |
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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
−8.059945930754727297595366959268, −7.32359072286218291491219127881, −6.55603325211088212268606381730, −6.43211355017728220097225468836, −5.09138519928181707408108350670, −4.28785798090436506647031547553, −3.79241505103862240279734556344, −2.40249983757529392465065132615, −1.19074058289614382677974888959, −0.097204747282683375743304357255,
1.67417065179895896293403150135, 2.73255033822680492965436508963, 3.09315973136029885392286308970, 4.34199930706385588743940477136, 5.05567215284813815854169122853, 5.75243948657010124363807065857, 6.33168586196092012126439168487, 7.76253563154113882053356247953, 8.348248283858231372849797407973, 9.012536462581593361671126697348
