| L(s) = 1 | + i·2-s − i·3-s − 4-s + (−2 − i)5-s + 6-s + 2i·7-s − i·8-s − 9-s + (1 − 2i)10-s + i·12-s − 2i·13-s − 2·14-s + (−1 + 2i)15-s + 16-s + 2i·17-s − i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (0.316 − 0.632i)10-s + 0.288i·12-s − 0.554i·13-s − 0.534·14-s + (−0.258 + 0.516i)15-s + 0.250·16-s + 0.485i·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.132801428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.132801428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2 + i)T \) |
| 37 | \( 1 + iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 10iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.687900147503881946912963157314, −8.929871208405485281868573329514, −8.061045071109727557833480120655, −7.74442124425856362625932106275, −6.68001892301503541499825680396, −5.79279481821316332130162607663, −5.03142468336652513445714619554, −3.93008288872750426353207224817, −2.77400149945224185587986444726, −1.04900108513294705733795950318,
0.64820559882020024916121422129, 2.50563471969673177083762867168, 3.52841878228554168905291947366, 4.24858544144779233391763768170, 4.99065158219781470986666377204, 6.41770390933790760297040453226, 7.29752072306438924333091167403, 8.168879983998667802800327860234, 9.002733054923657741675495751317, 9.949794568999077430049025979163
