| L(s) = 1 | + (1.31 + 0.0691i)2-s + (−0.985 − 1.21i)3-s + (−0.253 − 0.0266i)4-s + (−1.21 − 1.67i)6-s + (2.58 + 0.569i)7-s + (−2.94 − 0.465i)8-s + (0.113 − 0.536i)9-s + (0.468 − 0.0995i)11-s + (0.217 + 0.335i)12-s + (−0.195 − 0.0993i)13-s + (3.36 + 0.930i)14-s + (−3.34 − 0.712i)16-s + (−1.72 − 4.48i)17-s + (0.187 − 0.699i)18-s + (−0.561 − 5.33i)19-s + ⋯ |
| L(s) = 1 | + (0.932 + 0.0488i)2-s + (−0.568 − 0.702i)3-s + (−0.126 − 0.0133i)4-s + (−0.496 − 0.683i)6-s + (0.976 + 0.215i)7-s + (−1.04 − 0.164i)8-s + (0.0379 − 0.178i)9-s + (0.141 − 0.0300i)11-s + (0.0628 + 0.0967i)12-s + (−0.0540 − 0.0275i)13-s + (0.900 + 0.248i)14-s + (−0.837 − 0.178i)16-s + (−0.417 − 1.08i)17-s + (0.0441 − 0.164i)18-s + (−0.128 − 1.22i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.471 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.784217 - 1.30789i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.784217 - 1.30789i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + (-2.58 - 0.569i)T \) |
| good | 2 | \( 1 + (-1.31 - 0.0691i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (0.985 + 1.21i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-0.468 + 0.0995i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.195 + 0.0993i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (1.72 + 4.48i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.561 + 5.33i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.0278 + 0.532i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-2.22 + 3.06i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.36 + 3.05i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (6.39 - 4.15i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (2.67 - 0.870i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (6.03 + 6.03i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.98 + 2.68i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (3.76 - 3.04i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-3.58 + 3.97i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-6.61 + 5.95i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-12.1 + 4.65i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-3.80 - 2.76i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-7.14 + 11.0i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-3.76 - 8.46i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (1.48 - 9.38i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-3.91 - 4.35i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (1.50 + 9.51i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−9.815178648431229424898365158323, −8.982809195127308997662975611436, −8.126544017105066224134290871370, −6.91006294992871733077314289205, −6.41752173420437437936303397591, −5.18742548913469764801132301332, −4.84727479823566908495308254679, −3.59181240090333671877857433021, −2.25139718510748579697255341540, −0.58938630584483001777737322767,
1.83163411461813997424428498091, 3.53152513109967780085728318574, 4.28820638584397487993754669706, 5.02426505612079096894023912568, 5.66310907235190971111449996543, 6.68499124624763563002228207828, 8.061246870767386286784271537102, 8.622573430463747830734023422370, 9.846550432973491805606428757088, 10.53648761037209411000956006898
