| L(s) = 1 | − i·2-s + 1.14i·3-s − 4-s + 1.14·6-s + i·7-s + i·8-s + 1.68·9-s + 11-s − 1.14i·12-s + 4.68i·13-s + 14-s + 16-s + 0.292i·17-s − 1.68i·18-s − 6.51·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.661i·3-s − 0.5·4-s + 0.468·6-s + 0.377i·7-s + 0.353i·8-s + 0.561·9-s + 0.301·11-s − 0.330i·12-s + 1.29i·13-s + 0.267·14-s + 0.250·16-s + 0.0709i·17-s − 0.397i·18-s − 1.49·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.103159367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.103159367\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.14iT - 3T^{2} \) |
| 13 | \( 1 - 4.68iT - 13T^{2} \) |
| 17 | \( 1 - 0.292iT - 17T^{2} \) |
| 19 | \( 1 + 6.51T + 19T^{2} \) |
| 23 | \( 1 - 2.85iT - 23T^{2} \) |
| 29 | \( 1 - 1.43T + 29T^{2} \) |
| 31 | \( 1 - 0.978T + 31T^{2} \) |
| 37 | \( 1 + 0.853iT - 37T^{2} \) |
| 41 | \( 1 + 6.22T + 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 9.95iT - 47T^{2} \) |
| 53 | \( 1 - 5.43iT - 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 0.585iT - 67T^{2} \) |
| 71 | \( 1 + 0.335T + 71T^{2} \) |
| 73 | \( 1 + 3.70iT - 73T^{2} \) |
| 79 | \( 1 - 2.51T + 79T^{2} \) |
| 83 | \( 1 + 1.70iT - 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 9.10iT - 97T^{2} \) |
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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
−8.985928664059104573275732618477, −8.290345290014499579253619051539, −7.19748287053707383383875113651, −6.50897619102755239061428955997, −5.59648800414675926614151336206, −4.56662724385006830725981722769, −4.22443315260573073099002427706, −3.38317592375537067256259799492, −2.23040609210033764034906492342, −1.46966070622889518439787702656,
0.32553064783229775383035193167, 1.42917214168720239694966513693, 2.62107116414255730130743182485, 3.76452810123525042058964633821, 4.52429835875836522995834317342, 5.34171568496504230589790680687, 6.36570790029457207303476082007, 6.67530723317084969460257763917, 7.48339364485829945113023525048, 8.196496803660581389585884407889
