L(s) = 1 | + 2-s + 4-s + (−0.296 − 0.514i)5-s + 8-s + (−0.296 − 0.514i)10-s + (−0.296 + 0.514i)11-s + (1.25 − 2.17i)13-s + 16-s + (1.46 + 2.52i)17-s + (−2.69 + 4.66i)19-s + (−0.296 − 0.514i)20-s + (−0.296 + 0.514i)22-s + (2.23 + 3.86i)23-s + (2.32 − 4.02i)25-s + (1.25 − 2.17i)26-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.132 − 0.229i)5-s + 0.353·8-s + (−0.0938 − 0.162i)10-s + (−0.0894 + 0.154i)11-s + (0.348 − 0.603i)13-s + 0.250·16-s + (0.354 + 0.613i)17-s + (−0.617 + 1.06i)19-s + (−0.0663 − 0.114i)20-s + (−0.0632 + 0.109i)22-s + (0.465 + 0.805i)23-s + (0.464 − 0.804i)25-s + (0.246 − 0.427i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.941982840\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.941982840\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.296 + 0.514i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.296 - 0.514i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.25 + 2.17i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.46 - 2.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.69 - 4.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.23 - 3.86i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.09 - 5.36i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.136 - 0.236i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.58 + 9.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + (4.02 + 6.97i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 8.64T + 59T^{2} \) |
| 61 | \( 1 - 6.64T + 61T^{2} \) |
| 67 | \( 1 + 1.91T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + (3.95 + 6.85i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 9.24T + 79T^{2} \) |
| 83 | \( 1 + (-3.85 - 6.66i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.21 - 10.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.86 + 10.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.518062590069835633510404553624, −8.267738539062991788720122054117, −7.22459639420238937801023222110, −6.47334813469024795755583018753, −5.66884154340751149320796625122, −5.00371869805174516657987719008, −4.03005680824846716246603331893, −3.36041548160134776305202171843, −2.26831458500468797822487955636, −1.06739654025852017608160501244,
0.955530632807274432959694085230, 2.45390396917442011893584930961, 3.05756915219867755267484196444, 4.25134641261390697692885364757, 4.74409019519839262914896182191, 5.74849431061146314542889363268, 6.61068602576690115393557241720, 7.03748479963168972154137791546, 8.063778442612827181516734158755, 8.766320158142713517568447985350