| L(s) = 1 | + (3.84 − 2.78i)2-s + (−2.63 − 8.09i)3-s + (4.48 − 13.8i)4-s + (11.7 + 8.52i)5-s + (−32.6 − 23.7i)6-s + (−2.16 + 6.65i)7-s + (−9.57 − 29.4i)8-s + (−36.8 + 26.7i)9-s + 68.8·10-s + (34.2 + 12.5i)11-s − 123.·12-s + (−37.9 + 27.5i)13-s + (10.2 + 31.5i)14-s + (38.1 − 117. i)15-s + (−24.9 − 18.1i)16-s + (−68.7 − 49.9i)17-s + ⋯ |
| L(s) = 1 | + (1.35 − 0.986i)2-s + (−0.506 − 1.55i)3-s + (0.561 − 1.72i)4-s + (1.04 + 0.762i)5-s + (−2.22 − 1.61i)6-s + (−0.116 + 0.359i)7-s + (−0.423 − 1.30i)8-s + (−1.36 + 0.990i)9-s + 2.17·10-s + (0.938 + 0.344i)11-s − 2.97·12-s + (−0.809 + 0.588i)13-s + (0.196 + 0.603i)14-s + (0.657 − 2.02i)15-s + (−0.390 − 0.283i)16-s + (−0.980 − 0.712i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.20469 - 2.55470i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20469 - 2.55470i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.16 - 6.65i)T \) |
| 11 | \( 1 + (-34.2 - 12.5i)T \) |
| good | 2 | \( 1 + (-3.84 + 2.78i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (2.63 + 8.09i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-11.7 - 8.52i)T + (38.6 + 118. i)T^{2} \) |
| 13 | \( 1 + (37.9 - 27.5i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (68.7 + 49.9i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (4.03 + 12.4i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-41.4 + 127. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (171. - 124. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (46.4 - 142. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (56.2 + 173. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-141. - 435. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (427. - 310. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-137. + 423. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-317. - 230. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 421.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-282. - 205. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-212. + 654. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-854. + 620. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-426. - 309. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 416.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.13e3 - 821. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.46571837018791846871146570149, −12.54425680528128954359935790567, −11.75719078812322075437980207731, −10.90997020278938578785438524090, −9.392728170736338974703972476229, −6.93391387804967724428274037176, −6.35416914015655470317052627833, −5.01602551384957991990528531032, −2.68026485994796215983028359008, −1.70951799210008515051603233672,
3.65167516031952647536767338157, 4.83988653005925460339940337588, 5.51430033230117062284084423311, 6.70350887843358968615428211139, 8.820009678357150467808231408211, 9.865825886058926435175058230687, 11.14310399841903853669881240171, 12.61211378634307729352164169875, 13.45834045885920073955787485450, 14.67809360158105485717485796682
