| L(s) = 1 | + (0.730 + 0.155i)2-s + (−1.46 − 0.652i)3-s + (−1.31 − 0.586i)4-s + (0.802 − 2.08i)5-s + (−0.969 − 0.704i)6-s + (−2.45 + 0.993i)7-s + (−2.08 − 1.51i)8-s + (−0.284 − 0.315i)9-s + (0.910 − 1.40i)10-s + (−0.284 + 0.315i)11-s + (1.54 + 1.71i)12-s + (1.29 − 3.99i)13-s + (−1.94 + 0.344i)14-s + (−2.53 + 2.53i)15-s + (0.644 + 0.715i)16-s + (−0.114 + 1.08i)17-s + ⋯ |
| L(s) = 1 | + (0.516 + 0.109i)2-s + (−0.846 − 0.376i)3-s + (−0.658 − 0.293i)4-s + (0.358 − 0.933i)5-s + (−0.395 − 0.287i)6-s + (−0.926 + 0.375i)7-s + (−0.735 − 0.534i)8-s + (−0.0947 − 0.105i)9-s + (0.288 − 0.442i)10-s + (−0.0857 + 0.0952i)11-s + (0.446 + 0.496i)12-s + (0.360 − 1.10i)13-s + (−0.520 + 0.0921i)14-s + (−0.655 + 0.654i)15-s + (0.161 + 0.178i)16-s + (−0.0277 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.362829 - 0.638625i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.362829 - 0.638625i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.802 + 2.08i)T \) |
| 7 | \( 1 + (2.45 - 0.993i)T \) |
| good | 2 | \( 1 + (-0.730 - 0.155i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (1.46 + 0.652i)T + (2.00 + 2.22i)T^{2} \) |
| 11 | \( 1 + (0.284 - 0.315i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.29 + 3.99i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.114 - 1.08i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-7.39 + 3.29i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (2.87 + 0.610i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (-1.38 + 1.00i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.665 - 6.32i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-2.06 - 2.29i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-3.40 + 10.4i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 + (1.09 + 10.4i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-0.294 - 0.131i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-7.03 + 1.49i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (1.41 + 0.301i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-1.02 + 9.76i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-6.56 + 4.76i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (5.37 - 5.97i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-1.29 - 12.3i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-2.81 - 2.04i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (2.78 + 0.591i)T + (81.3 + 36.1i)T^{2} \) |
| 97 | \( 1 + (-5.06 + 3.67i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.50907600789062235953188972696, −11.86394831547964054731492679643, −10.27666504357968840834245255817, −9.398820952767693908012268389828, −8.469115967517642421079723535274, −6.69340720207206062146716533704, −5.62511012590711046966477215708, −5.19897481100132432891156809520, −3.43100371486225664944793069788, −0.65472839181342999879916847129,
3.03923797834126251638494176788, 4.20197926848244787027319244326, 5.58597981110548356336024280362, 6.38491411736507592277351270849, 7.76613159067916080879059601720, 9.435993091702132276434971472968, 10.02108343143169258999829782693, 11.30950221799563008659869251030, 11.87333044058168365897850330860, 13.23238139847612602413290358818
