| L(s) = 1 | + 2-s + 4-s + (−1.93 + 1.80i)7-s + 8-s − 3.87i·11-s − 1.60·13-s + (−1.93 + 1.80i)14-s + 16-s − 8.11i·17-s + 2.63i·19-s − 3.87i·22-s + 5.47·23-s − 1.60·26-s + (−1.93 + 1.80i)28-s + 5.47i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.732 + 0.681i)7-s + 0.353·8-s − 1.16i·11-s − 0.445·13-s + (−0.517 + 0.481i)14-s + 0.250·16-s − 1.96i·17-s + 0.605i·19-s − 0.825i·22-s + 1.14·23-s − 0.314·26-s + (−0.366 + 0.340i)28-s + 1.01i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.158785871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.158785871\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.93 - 1.80i)T \) |
| good | 11 | \( 1 + 3.87iT - 11T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 17 | \( 1 + 8.11iT - 17T^{2} \) |
| 19 | \( 1 - 2.63iT - 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 - 5.47iT - 29T^{2} \) |
| 31 | \( 1 - 3.73iT - 31T^{2} \) |
| 37 | \( 1 + 4.51iT - 37T^{2} \) |
| 41 | \( 1 + 1.60T + 41T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 11.1iT - 47T^{2} \) |
| 53 | \( 1 + 2.26T + 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 + 11.8iT - 61T^{2} \) |
| 67 | \( 1 + 6.90iT - 67T^{2} \) |
| 71 | \( 1 + 2.63iT - 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 8.01T + 79T^{2} \) |
| 83 | \( 1 - 3.20iT - 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 8.68T + 97T^{2} \) |
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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
−8.755330888268550120684744140990, −7.53561089714368482105030007568, −6.93112714161836962573004115022, −6.19381481933793898995378151911, −5.27534644090836637046615164074, −4.98086289277623844575653016572, −3.45997565510877550546380971304, −3.15404008744243660819514093101, −2.13415623329750287609864253305, −0.52444824243644774132262897061,
1.29817563836353451265011282374, 2.45725995683282654793768652390, 3.34783456709008293357639479233, 4.34929652012104093354322397673, 4.67174352543235743122995192541, 5.98690874817926186193061902270, 6.42119224604955308847526252258, 7.31264907046808807303689876575, 7.78380728106547524112998102443, 8.880578845338611870090868150532
