| L(s) = 1 | + 5i·7-s − 14·11-s − i·13-s − 46i·17-s − 19·19-s − 46i·23-s + 14·29-s + 133·31-s + 258i·37-s − 84·41-s + 167i·43-s − 410i·47-s + 318·49-s + 456i·53-s − 194·59-s + ⋯ |
| L(s) = 1 | + 0.269i·7-s − 0.383·11-s − 0.0213i·13-s − 0.656i·17-s − 0.229·19-s − 0.417i·23-s + 0.0896·29-s + 0.770·31-s + 1.14i·37-s − 0.319·41-s + 0.592i·43-s − 1.27i·47-s + 0.927·49-s + 1.18i·53-s − 0.428·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.006866187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.006866187\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5iT - 343T^{2} \) |
| 11 | \( 1 + 14T + 1.33e3T^{2} \) |
| 13 | \( 1 + iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 46iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 19T + 6.85e3T^{2} \) |
| 23 | \( 1 + 46iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 14T + 2.43e4T^{2} \) |
| 31 | \( 1 - 133T + 2.97e4T^{2} \) |
| 37 | \( 1 - 258iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 84T + 6.89e4T^{2} \) |
| 43 | \( 1 - 167iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 410iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 456iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 194T + 2.05e5T^{2} \) |
| 61 | \( 1 + 17T + 2.26e5T^{2} \) |
| 67 | \( 1 - 653iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 828T + 3.57e5T^{2} \) |
| 73 | \( 1 + 570iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 552T + 4.93e5T^{2} \) |
| 83 | \( 1 - 142iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 841iT - 9.12e5T^{2} \) |
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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
−9.082133900033455591715144742662, −8.433450826693909415053468609062, −7.62286755274128728592367266603, −6.78660211770914979109623115383, −5.97713656241324522250851016807, −5.07628384929594705147211156638, −4.32212075797690691869831929344, −3.12784551227140321830914791744, −2.34057615370752309759200190601, −1.03859485927879428294446752631,
0.23136645742809323516223790514, 1.50150977452343665718071761984, 2.59410291728196429053821993872, 3.65745635718767971466762032472, 4.48464602346167380237537251035, 5.45846272535351158481225122956, 6.23860887190096125143912679361, 7.13212757852434674979840553204, 7.88061089006616393460637474878, 8.617190674675728636225261170691
