| L(s) = 1 | − 6.92i·3-s + 9.10i·7-s − 20.9·9-s − 10.6·11-s − 63.6i·13-s + 77.2i·17-s − 19·19-s + 63.0·21-s − 144. i·23-s − 42.2i·27-s − 51.2·29-s + 71.5·31-s + 73.3i·33-s − 75.9i·37-s − 440.·39-s + ⋯ |
| L(s) = 1 | − 1.33i·3-s + 0.491i·7-s − 0.774·9-s − 0.290·11-s − 1.35i·13-s + 1.10i·17-s − 0.229·19-s + 0.655·21-s − 1.31i·23-s − 0.300i·27-s − 0.327·29-s + 0.414·31-s + 0.387i·33-s − 0.337i·37-s − 1.80·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3802842626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3802842626\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 6.92iT - 27T^{2} \) |
| 7 | \( 1 - 9.10iT - 343T^{2} \) |
| 11 | \( 1 + 10.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 63.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 77.2iT - 4.91e3T^{2} \) |
| 23 | \( 1 + 144. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 51.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 71.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 75.9iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 1.95T + 6.89e4T^{2} \) |
| 43 | \( 1 - 242. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 594. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 124. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 556.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 0.544T + 2.26e5T^{2} \) |
| 67 | \( 1 - 634. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 163.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 428. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 903. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 241.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 10.8iT - 9.12e5T^{2} \) |
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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
−8.284250485349568429214175861551, −7.59402582442255682300365089449, −6.74706046822283347571043700116, −6.02724028480045338807911644372, −5.37524601932565133278717217985, −4.14733282097920278889262072946, −2.90093053329723771350918442657, −2.16827425089326198400222730124, −1.10420348503545236004057240920, −0.082257819260734454519958181223,
1.48318600099407123281576413023, 2.82991509025072895617553702576, 3.77481472285331268819718796455, 4.47754614709782764300673622038, 5.08536785692556078148392901012, 6.10333403590739738515687311572, 7.11949005260759134022725332698, 7.76107209465206607620662628649, 9.122185229598800280355790326059, 9.232652923066758701139921410209
