| L(s) = 1 | + (0.453 + 0.891i)2-s + (0.256 − 1.61i)3-s + (−0.587 + 0.809i)4-s + (1.69 − 1.45i)5-s + (1.55 − 0.506i)6-s + (−0.753 + 0.119i)7-s + (−0.987 − 0.156i)8-s + (0.299 + 0.0972i)9-s + (2.06 + 0.849i)10-s + (−1.49 + 2.95i)11-s + (1.15 + 1.15i)12-s + (−0.760 + 0.387i)13-s + (−0.448 − 0.616i)14-s + (−1.92 − 3.11i)15-s + (−0.309 − 0.951i)16-s + (−1.29 − 0.657i)17-s + ⋯ |
| L(s) = 1 | + (0.321 + 0.630i)2-s + (0.148 − 0.934i)3-s + (−0.293 + 0.404i)4-s + (0.758 − 0.651i)5-s + (0.636 − 0.206i)6-s + (−0.284 + 0.0450i)7-s + (−0.349 − 0.0553i)8-s + (0.0997 + 0.0324i)9-s + (0.654 + 0.268i)10-s + (−0.451 + 0.892i)11-s + (0.334 + 0.334i)12-s + (−0.210 + 0.107i)13-s + (−0.119 − 0.164i)14-s + (−0.496 − 0.805i)15-s + (−0.0772 − 0.237i)16-s + (−0.312 − 0.159i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28045 - 0.00807911i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28045 - 0.00807911i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.453 - 0.891i)T \) |
| 5 | \( 1 + (-1.69 + 1.45i)T \) |
| 11 | \( 1 + (1.49 - 2.95i)T \) |
| good | 3 | \( 1 + (-0.256 + 1.61i)T + (-2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (0.753 - 0.119i)T + (6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (0.760 - 0.387i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (1.29 + 0.657i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (4.09 - 2.97i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.65 - 1.65i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.552 + 0.401i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.08 - 3.34i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.26 + 8.01i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-7.30 - 10.0i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (7.61 + 7.61i)T + 43iT^{2} \) |
| 47 | \( 1 + (-12.5 - 1.98i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-3.72 - 7.31i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-0.254 + 0.350i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-12.0 + 3.91i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (3.82 + 3.82i)T + 67iT^{2} \) |
| 71 | \( 1 + (2.45 + 7.55i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.179 + 1.13i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-3.11 + 9.59i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.108 - 0.212i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + 11.2iT - 89T^{2} \) |
| 97 | \( 1 + (12.6 - 6.45i)T + (57.0 - 78.4i)T^{2} \) |
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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
−13.53185756604076712860674059886, −12.77380566848914963920251571759, −12.25673806352418367185086026774, −10.31302828739094360696929116096, −9.185242642040558651579534174604, −7.966776030448576643360422712566, −6.95354935087037357364341338520, −5.85068525115254047686373548393, −4.48865527530951970505472553136, −2.08852039325828745864020529314,
2.63027926661264606270276793065, 3.97186453181384143630987850519, 5.41809437894301104002144961976, 6.70461595636324576281937467807, 8.669233118919703737310699810902, 9.771494551860266179334534607667, 10.47706678833000787961313027160, 11.25501212753385586927680185096, 12.83174449728444091674432520937, 13.59517584398137496180490933772
