| L(s) = 1 | + (0.295 + 0.811i)2-s + (8.27 + 3.01i)3-s + (5.55 − 4.66i)4-s + (−17.3 + 3.06i)5-s + 7.60i·6-s + (−2.43 − 13.8i)7-s + (11.4 + 6.58i)8-s + (38.6 + 32.4i)9-s + (−7.62 − 13.2i)10-s + (−20.3 + 35.2i)11-s + (60.0 − 21.8i)12-s + (−40.3 − 48.0i)13-s + (10.5 − 6.06i)14-s + (−153. − 27.0i)15-s + (8.10 − 45.9i)16-s + (31.3 − 37.3i)17-s + ⋯ |
| L(s) = 1 | + (0.104 + 0.286i)2-s + (1.59 + 0.579i)3-s + (0.694 − 0.582i)4-s + (−1.55 + 0.274i)5-s + 0.517i·6-s + (−0.131 − 0.747i)7-s + (0.504 + 0.291i)8-s + (1.43 + 1.20i)9-s + (−0.241 − 0.417i)10-s + (−0.558 + 0.967i)11-s + (1.44 − 0.525i)12-s + (−0.860 − 1.02i)13-s + (0.200 − 0.115i)14-s + (−2.63 − 0.464i)15-s + (0.126 − 0.717i)16-s + (0.447 − 0.533i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.80171 + 0.452921i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80171 + 0.452921i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-192. - 116. i)T \) |
| good | 2 | \( 1 + (-0.295 - 0.811i)T + (-6.12 + 5.14i)T^{2} \) |
| 3 | \( 1 + (-8.27 - 3.01i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (17.3 - 3.06i)T + (117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (2.43 + 13.8i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (20.3 - 35.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (40.3 + 48.0i)T + (-381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-31.3 + 37.3i)T + (-853. - 4.83e3i)T^{2} \) |
| 19 | \( 1 + (26.9 - 74.1i)T + (-5.25e3 - 4.40e3i)T^{2} \) |
| 23 | \( 1 + (-42.7 + 24.7i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (110. + 63.9i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 108. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-148. + 124. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 - 333. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-100. - 173. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (106. - 605. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (326. + 57.5i)T + (1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (93.5 + 111. i)T + (-3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (135. + 768. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-150. - 54.6i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 + 178.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-662. + 116. i)T + (4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (172. + 144. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (1.16e3 + 206. i)T + (6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-529. + 305. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.42782284597225912440253726463, −15.08792699575049440263454781481, −14.17913177515365182631586621153, −12.49707351552757455817024075623, −10.75326817332864303914054438212, −9.828237824627021676299187934561, −7.75813349808751591914152921135, −7.50618754284899687177911732595, −4.49844130513987675717358937052, −2.94586312275028110397143284639,
2.55793075278331469479274921440, 3.81618331441020336993114331796, 7.17088502160360521818591470757, 8.043167803146434105264352878910, 8.974910460631716278550607221794, 11.30410288571214888153110949256, 12.30977973378247048523979301206, 13.20431600045511622332581105350, 14.78117936548683629896271590326, 15.56319117765530594347747731235
