| L(s) = 1 | − 5.26i·3-s + 12.1·7-s − 18.7·9-s − 4.59·11-s − 19.9i·13-s − 4.10·17-s + (−17.9 − 6.30i)19-s − 64.0i·21-s + 35.2·23-s + 51.1i·27-s + 16.4i·29-s + 4.01i·31-s + 24.2i·33-s − 10.6i·37-s − 104.·39-s + ⋯ |
| L(s) = 1 | − 1.75i·3-s + 1.73·7-s − 2.08·9-s − 0.417·11-s − 1.53i·13-s − 0.241·17-s + (−0.943 − 0.331i)19-s − 3.05i·21-s + 1.53·23-s + 1.89i·27-s + 0.568i·29-s + 0.129i·31-s + 0.733i·33-s − 0.286i·37-s − 2.68·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.703189216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.703189216\) |
| \(L(2)\) |
|
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 | \( 1 + (17.9 + 6.30i)T \) |
| good | 3 | \( 1 + 5.26iT - 9T^{2} \) |
| 7 | \( 1 - 12.1T + 49T^{2} \) |
| 11 | \( 1 + 4.59T + 121T^{2} \) |
| 13 | \( 1 + 19.9iT - 169T^{2} \) |
| 17 | \( 1 + 4.10T + 289T^{2} \) |
| 23 | \( 1 - 35.2T + 529T^{2} \) |
| 29 | \( 1 - 16.4iT - 841T^{2} \) |
| 31 | \( 1 - 4.01iT - 961T^{2} \) |
| 37 | \( 1 + 10.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 22.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 23.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 97.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 75.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 6.31iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 118. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7.02T + 5.32e3T^{2} \) |
| 79 | \( 1 + 76.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 100.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 123. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 101. iT - 9.40e3T^{2} \) |
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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
−8.365442650368916456485953622303, −7.78601062388293954135820538182, −7.21541094489128450048230878431, −6.36180285226901694151846698860, −5.33967278816591224518342583568, −4.87177303093680957591263170428, −3.18067047617613907635967893612, −2.21952977786173597375821844339, −1.41581987829125621319202036822, −0.42656808163246210597556904399,
1.62347921036801094580860453780, 2.76752638763144829499444066041, 4.03797380223017494945341190080, 4.59920484331720111524513945582, 5.01624690356686142391909231701, 6.03252993740488944448379812207, 7.18194324506313732700055213492, 8.296081818003240117751926944214, 8.709742727972477057099774496112, 9.452559076797879678588081855604
