| L(s) = 1 | + (−0.0374 + 0.00196i)2-s + (0.0687 − 0.0849i)3-s + (−1.98 + 0.208i)4-s + (−0.633 + 2.14i)5-s + (−0.00240 + 0.00331i)6-s + (−2.20 + 1.46i)7-s + (0.147 − 0.0234i)8-s + (0.621 + 2.92i)9-s + (0.0195 − 0.0814i)10-s + (−1.34 − 0.286i)11-s + (−0.118 + 0.183i)12-s + (1.43 − 0.730i)13-s + (0.0795 − 0.0591i)14-s + (0.138 + 0.201i)15-s + (3.90 − 0.829i)16-s + (−1.56 + 4.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.0264 + 0.00138i)2-s + (0.0397 − 0.0490i)3-s + (−0.993 + 0.104i)4-s + (−0.283 + 0.959i)5-s + (−0.000982 + 0.00135i)6-s + (−0.832 + 0.553i)7-s + (0.0523 − 0.00828i)8-s + (0.207 + 0.974i)9-s + (0.00616 − 0.0257i)10-s + (−0.405 − 0.0862i)11-s + (−0.0343 + 0.0528i)12-s + (0.397 − 0.202i)13-s + (0.0212 − 0.0158i)14-s + (0.0357 + 0.0519i)15-s + (0.976 − 0.207i)16-s + (−0.379 + 0.988i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.363124 + 0.554380i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.363124 + 0.554380i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.633 - 2.14i)T \) |
| 7 | \( 1 + (2.20 - 1.46i)T \) |
| good | 2 | \( 1 + (0.0374 - 0.00196i)T + (1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (-0.0687 + 0.0849i)T + (-0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (1.34 + 0.286i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.43 + 0.730i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (1.56 - 4.07i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (0.00716 - 0.0681i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (0.295 + 5.63i)T + (-22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (-3.88 - 5.35i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.42 - 3.20i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (6.49 + 4.21i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (-9.12 - 2.96i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (3.38 - 3.38i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.65 + 2.55i)T + (34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (-2.30 - 1.86i)T + (11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-6.88 - 7.64i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-5.62 - 5.06i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (8.28 + 3.18i)T + (49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (-5.52 + 4.01i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.233 + 0.359i)T + (-29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (3.70 - 8.31i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.663 - 4.19i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (5.38 - 5.98i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (2.59 - 16.3i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−13.03569777353592976958235360381, −12.29543031812082675078667323581, −10.69347390343842009443653002934, −10.27321365907233995003572647732, −8.863510950414942924213547324843, −8.065082955084450673570878848469, −6.76310603703295824664423613654, −5.53201649285790478084630093346, −4.08519572112683341007534737314, −2.74816327879635601932379705399,
0.64254826302408406250477063436, 3.60048323489881482859181013965, 4.53897413015696328973454777320, 5.84211148937111581582984268644, 7.28469081363620519026147377990, 8.556655442421699999116852303421, 9.418201583989384896100347577722, 10.00595760663901262319690503297, 11.61963013928648844441099675043, 12.59041458733463643186447079594
