| L(s) = 1 | + 0.278·3-s − 2.23i·5-s − 8.51i·7-s − 8.92·9-s + 7.57i·11-s − 2.64·13-s − 0.622i·15-s + 7.56i·17-s − 24.2i·19-s − 2.37i·21-s + (15.7 − 16.7i)23-s − 5.00·25-s − 4.99·27-s − 31.8·29-s + 56.5·31-s + ⋯ |
| L(s) = 1 | + 0.0928·3-s − 0.447i·5-s − 1.21i·7-s − 0.991·9-s + 0.688i·11-s − 0.203·13-s − 0.0415i·15-s + 0.444i·17-s − 1.27i·19-s − 0.112i·21-s + (0.684 − 0.729i)23-s − 0.200·25-s − 0.184·27-s − 1.09·29-s + 1.82·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04234591365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04234591365\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-15.7 + 16.7i)T \) |
| good | 3 | \( 1 - 0.278T + 9T^{2} \) |
| 7 | \( 1 + 8.51iT - 49T^{2} \) |
| 11 | \( 1 - 7.57iT - 121T^{2} \) |
| 13 | \( 1 + 2.64T + 169T^{2} \) |
| 17 | \( 1 - 7.56iT - 289T^{2} \) |
| 19 | \( 1 + 24.2iT - 361T^{2} \) |
| 29 | \( 1 + 31.8T + 841T^{2} \) |
| 31 | \( 1 - 56.5T + 961T^{2} \) |
| 37 | \( 1 - 39.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 42.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 84.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 11.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 67.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 35.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 44.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 8.86T + 5.04e3T^{2} \) |
| 73 | \( 1 + 87.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 154. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 141. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 63.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 143. iT - 9.40e3T^{2} \) |
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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
−8.509302205616725447748204482643, −7.892710414622887650442091280802, −6.94375617932163949426798132115, −6.36762652256798667751134182517, −5.05016514477686829545187151924, −4.59907731066928153446204079254, −3.50948163739661449174483705939, −2.50609653142304299979836046155, −1.16129295393907946161641365386, −0.01113508585788618775112025097,
1.76237847281188859745311873383, 2.88397245197431873212672281373, 3.36531676528793139561189479488, 4.81181456993647614030000808782, 5.81968742779450406709071801181, 6.02280585104150733228461766023, 7.25387475451118679637995800769, 8.147218575888742175255911861737, 8.710354968985805932499916359828, 9.450670441447505962906323434718
