| L(s) = 1 | + (1.59 − 0.428i)2-s + (12.4 − 7.17i)3-s + (−11.4 + 6.62i)4-s + (32.1 + 8.60i)5-s + (16.8 − 16.8i)6-s + (−20.2 − 35.0i)7-s + (−34.2 + 34.2i)8-s + (62.5 − 108. i)9-s + 55.0·10-s + 10.5i·11-s + (−95.1 + 164. i)12-s + (−38.1 − 10.2i)13-s + (−47.4 − 47.4i)14-s + (461. − 123. i)15-s + (65.9 − 114. i)16-s + (138. + 515. i)17-s + ⋯ |
| L(s) = 1 | + (0.399 − 0.107i)2-s + (1.38 − 0.797i)3-s + (−0.717 + 0.414i)4-s + (1.28 + 0.344i)5-s + (0.466 − 0.466i)6-s + (−0.413 − 0.716i)7-s + (−0.535 + 0.535i)8-s + (0.772 − 1.33i)9-s + 0.550·10-s + 0.0874i·11-s + (−0.660 + 1.14i)12-s + (−0.225 − 0.0604i)13-s + (−0.242 − 0.242i)14-s + (2.05 − 0.549i)15-s + (0.257 − 0.446i)16-s + (0.478 + 1.78i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.34517 - 0.582813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.34517 - 0.582813i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-994. + 940. i)T \) |
| good | 2 | \( 1 + (-1.59 + 0.428i)T + (13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (-12.4 + 7.17i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-32.1 - 8.60i)T + (541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (20.2 + 35.0i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 10.5iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (38.1 + 10.2i)T + (2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-138. - 515. i)T + (-7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (643. + 172. i)T + (1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (362. - 362. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-19.5 - 19.5i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (-113. - 113. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (-2.57e3 + 1.48e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.32e3 - 1.32e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + 279.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-909. + 1.57e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (1.25e3 + 4.67e3i)T + (-1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (543. - 2.03e3i)T + (-1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-2.10e3 + 1.21e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-2.01e3 - 3.48e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + 6.52e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (8.83e3 + 2.36e3i)T + (3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-1.81e3 + 3.13e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-6.84e3 + 1.83e3i)T + (5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (2.33e3 - 2.33e3i)T - 8.85e7iT^{2} \) |
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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
−14.77966205777797555633021076491, −14.16739864069812222015374207442, −13.14341761926465193707154938744, −12.80436327560092625676113858124, −10.25412622475904144524078253563, −9.063571698348228783633074947047, −7.88357889380628462004122248327, −6.25546739032850273113213556783, −3.79035370512824171659444469473, −2.17459148712245466389505057035,
2.58825716978231881704100662625, 4.50577631793053638489763611391, 5.94780806058035299486356551500, 8.575209124198479156402568935724, 9.452739830489519661778492825968, 10.04393743730744247764839080854, 12.66543408394607760489012758153, 13.71979826670136934686647341688, 14.37844570689983500382325687469, 15.32212824468711688981546639068
