| L(s) = 1 | + (1 − i)5-s − i·9-s − 17-s − i·25-s + (−1 + i)29-s + (1 − i)37-s + (1 + i)41-s + (−1 − i)45-s + i·49-s + (1 + i)61-s + (1 − i)73-s − 81-s + (−1 + i)85-s + (−1 + i)97-s + (1 + i)109-s + ⋯ |
| L(s) = 1 | + (1 − i)5-s − i·9-s − 17-s − i·25-s + (−1 + i)29-s + (1 − i)37-s + (1 + i)41-s + (−1 − i)45-s + i·49-s + (1 + i)61-s + (1 − i)73-s − 81-s + (−1 + i)85-s + (−1 + i)97-s + (1 + i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.179720680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.179720680\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + iT^{2} \) |
| 5 | \( 1 + (-1 + i)T - iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (1 - i)T - iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + iT^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-1 - i)T + iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.588934908138831520673642830554, −9.265571881244834337357579965470, −8.591015462312361533047001540641, −7.43684079776867958719097700106, −6.39767982892854873483723283353, −5.75348249622137589163202892565, −4.80935170246932319352528821193, −3.85040878318831038092186543401, −2.44500032978931932182990133389, −1.19958702660243868730913749486,
2.01372825311750430550157793700, 2.63719026036319628671336645673, 4.02410084025536625294428768304, 5.16479498597968938008930051763, 6.02050679958439752189492684587, 6.78071919844583122582614699554, 7.61972907446342808951921247383, 8.545928972984659761597550506437, 9.603724719281053862515834714837, 10.12389861972819212961232711084
