| L(s) = 1 | − 2.24·3-s − i·5-s − 1.52·7-s + 2.05·9-s − 2.71·11-s − 0.941i·13-s + 2.24i·15-s + 4.83i·17-s − 0.249i·19-s + 3.43·21-s + 0.941i·23-s − 25-s + 2.11·27-s − 0.719i·29-s + 4.02i·31-s + ⋯ |
| L(s) = 1 | − 1.29·3-s − 0.447i·5-s − 0.578·7-s + 0.686·9-s − 0.820·11-s − 0.261i·13-s + 0.580i·15-s + 1.17i·17-s − 0.0571i·19-s + 0.750·21-s + 0.196i·23-s − 0.200·25-s + 0.407·27-s − 0.133i·29-s + 0.723i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5742151099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5742151099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 37 | \( 1 + (4.71 - 3.83i)T \) |
| good | 3 | \( 1 + 2.24T + 3T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 + 2.71T + 11T^{2} \) |
| 13 | \( 1 + 0.941iT - 13T^{2} \) |
| 17 | \( 1 - 4.83iT - 17T^{2} \) |
| 19 | \( 1 + 0.249iT - 19T^{2} \) |
| 23 | \( 1 - 0.941iT - 23T^{2} \) |
| 29 | \( 1 + 0.719iT - 29T^{2} \) |
| 31 | \( 1 - 4.02iT - 31T^{2} \) |
| 41 | \( 1 + 8.27T + 41T^{2} \) |
| 43 | \( 1 - 2.71iT - 43T^{2} \) |
| 47 | \( 1 - 3.30T + 47T^{2} \) |
| 53 | \( 1 - 8.39T + 53T^{2} \) |
| 59 | \( 1 + 7.30iT - 59T^{2} \) |
| 61 | \( 1 + 6.83iT - 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 - 3.05T + 71T^{2} \) |
| 73 | \( 1 - 9.11T + 73T^{2} \) |
| 79 | \( 1 + 1.75iT - 79T^{2} \) |
| 83 | \( 1 + 0.131T + 83T^{2} \) |
| 89 | \( 1 - 8.99iT - 89T^{2} \) |
| 97 | \( 1 + 16.2iT - 97T^{2} \) |
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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
−8.500884421368223777027874672633, −7.986554229226979325612998265918, −6.80408425073184866672988627962, −6.38173995711418767073951327621, −5.37896271113428932029840588169, −5.14803263583355165294757814999, −4.01567973891259597427615188209, −3.03840911195160497057671030185, −1.66629943853898601097681289229, −0.37488373684976635258889377929,
0.62591051717225682705630857296, 2.26815359008776403317382824981, 3.20093776211645625914491208397, 4.29185264532530439110863735373, 5.28352348266811809265914173274, 5.64477168035966179959375521113, 6.69498763969642364913650373882, 6.99740361210097991327936358735, 7.959173510206461080277541005993, 8.952183310967372196602021890359
