| L(s) = 1 | + (0.998 − 0.0475i)2-s + (1.50 − 0.850i)3-s + (0.995 − 0.0950i)4-s + (0.631 + 0.601i)5-s + (1.46 − 0.921i)6-s + (0.849 + 4.40i)7-s + (0.989 − 0.142i)8-s + (1.55 − 2.56i)9-s + (0.659 + 0.571i)10-s + (0.387 − 0.199i)11-s + (1.42 − 0.989i)12-s + (−5.47 − 1.05i)13-s + (1.05 + 4.36i)14-s + (1.46 + 0.371i)15-s + (0.981 − 0.189i)16-s + (0.134 + 0.295i)17-s + ⋯ |
| L(s) = 1 | + (0.706 − 0.0336i)2-s + (0.871 − 0.490i)3-s + (0.497 − 0.0475i)4-s + (0.282 + 0.269i)5-s + (0.598 − 0.376i)6-s + (0.321 + 1.66i)7-s + (0.349 − 0.0503i)8-s + (0.518 − 0.855i)9-s + (0.208 + 0.180i)10-s + (0.116 − 0.0602i)11-s + (0.410 − 0.285i)12-s + (−1.51 − 0.292i)13-s + (0.282 + 1.16i)14-s + (0.378 + 0.0959i)15-s + (0.245 − 0.0473i)16-s + (0.0326 + 0.0715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.75650 - 0.0393086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.75650 - 0.0393086i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.998 + 0.0475i)T \) |
| 3 | \( 1 + (-1.50 + 0.850i)T \) |
| 23 | \( 1 + (-0.486 - 4.77i)T \) |
| good | 5 | \( 1 + (-0.631 - 0.601i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.849 - 4.40i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (-0.387 + 0.199i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (5.47 + 1.05i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-0.134 - 0.295i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (5.74 + 2.62i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.750 + 7.86i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (-2.09 + 1.64i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (-1.31 + 4.48i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (5.29 - 5.54i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-7.50 + 9.53i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (3.75 + 2.16i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.70 - 6.57i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (0.803 - 4.16i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (3.56 - 8.89i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (2.01 - 3.90i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (-6.80 - 10.5i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.190 + 0.417i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (4.13 - 1.43i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-0.753 + 0.718i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (1.48 - 10.3i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.991 - 0.240i)T + (86.2 + 44.4i)T^{2} \) |
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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
−11.67067831080648107045466437741, −10.22296210038340672737393154669, −9.305675168487973549296209316887, −8.434512912614777652262482860620, −7.48568390320152218870674090100, −6.39649761538384003722456568664, −5.51185952032919103690489810799, −4.26487290764249245095203327953, −2.58742685059584877968433635299, −2.29531514243024567696800424399,
1.83692349592002959530753275485, 3.29639846234176971851457099972, 4.43827860927224643951922410347, 4.89135701370363439749291601927, 6.68743270812760000989212628400, 7.44404699268230381169818249580, 8.365474407351315161300907459136, 9.587616813887639877465513399769, 10.36777104209645028855658362092, 10.98836817179259449049734429050
