| L(s) = 1 | + (−2.77 − 0.145i)2-s + (0.835 + 1.03i)3-s + (5.68 + 0.597i)4-s + (0.745 + 2.10i)5-s + (−2.16 − 2.98i)6-s + (2.14 + 1.54i)7-s + (−10.2 − 1.61i)8-s + (0.257 − 1.21i)9-s + (−1.76 − 5.95i)10-s + (0.0303 − 0.00645i)11-s + (4.13 + 6.36i)12-s + (−2.44 − 1.24i)13-s + (−5.73 − 4.60i)14-s + (−1.55 + 2.52i)15-s + (16.9 + 3.59i)16-s + (1.52 + 3.97i)17-s + ⋯ |
| L(s) = 1 | + (−1.96 − 0.102i)2-s + (0.482 + 0.595i)3-s + (2.84 + 0.298i)4-s + (0.333 + 0.942i)5-s + (−0.884 − 1.21i)6-s + (0.811 + 0.584i)7-s + (−3.61 − 0.571i)8-s + (0.0859 − 0.404i)9-s + (−0.557 − 1.88i)10-s + (0.00916 − 0.00194i)11-s + (1.19 + 1.83i)12-s + (−0.676 − 0.344i)13-s + (−1.53 − 1.23i)14-s + (−0.400 + 0.653i)15-s + (4.22 + 0.898i)16-s + (0.369 + 0.963i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.534813 + 0.376902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.534813 + 0.376902i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.745 - 2.10i)T \) |
| 7 | \( 1 + (-2.14 - 1.54i)T \) |
| good | 2 | \( 1 + (2.77 + 0.145i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (-0.835 - 1.03i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-0.0303 + 0.00645i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (2.44 + 1.24i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.52 - 3.97i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.275 - 2.62i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.103 + 1.97i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-0.472 + 0.649i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.468 - 1.05i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (0.394 - 0.255i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (3.35 - 1.08i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (6.90 + 6.90i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.24 - 1.24i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-8.21 + 6.64i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (1.58 - 1.76i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.651 + 0.586i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-9.68 + 3.71i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (2.26 + 1.64i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.90 - 6.01i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (4.45 + 10.0i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-2.29 + 14.4i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (3.12 + 3.47i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (0.0409 + 0.258i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−12.33446291272637355374742878264, −11.50478799019058739050846016853, −10.34513937777781560078366294062, −10.03898966374833192484844474056, −8.894122801912340820023780169833, −8.148345980217556698073065735780, −7.07050655643058493717124402762, −5.90609638434582329155024536074, −3.27287572252093809388562587666, −1.99979752727351560993746114426,
1.16776903696735866588507916619, 2.35104441945601269227298936720, 5.17496070417930013042731893805, 6.93373545533882730342917641387, 7.66170093115102767857877659012, 8.422125800463690273469651827465, 9.319920967855653799180555994500, 10.19126641225114426706819427904, 11.31836680666294505633080160934, 12.15316120756119363412196105777
