| L(s) = 1 | + (−0.429 − 2.70i)2-s + (0.821 − 1.61i)3-s + (−5.25 + 1.70i)4-s + (−1.80 − 1.32i)5-s + (−4.71 − 1.53i)6-s + (2.44 − 1.01i)7-s + (4.38 + 8.60i)8-s + (−0.161 − 0.222i)9-s + (−2.82 + 5.44i)10-s + (−1.02 − 0.747i)11-s + (−1.56 + 9.86i)12-s + (2.36 + 0.374i)13-s + (−3.79 − 6.18i)14-s + (−3.61 + 1.81i)15-s + (12.4 − 9.07i)16-s + (−2.60 − 5.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.303 − 1.91i)2-s + (0.474 − 0.930i)3-s + (−2.62 + 0.853i)4-s + (−0.805 − 0.593i)5-s + (−1.92 − 0.626i)6-s + (0.923 − 0.383i)7-s + (1.54 + 3.04i)8-s + (−0.0538 − 0.0741i)9-s + (−0.891 + 1.72i)10-s + (−0.310 − 0.225i)11-s + (−0.451 + 2.84i)12-s + (0.655 + 0.103i)13-s + (−1.01 − 1.65i)14-s + (−0.934 + 0.468i)15-s + (3.12 − 2.26i)16-s + (−0.632 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.303555 + 0.848006i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.303555 + 0.848006i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.80 + 1.32i)T \) |
| 7 | \( 1 + (-2.44 + 1.01i)T \) |
| good | 2 | \( 1 + (0.429 + 2.70i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (-0.821 + 1.61i)T + (-1.76 - 2.42i)T^{2} \) |
| 11 | \( 1 + (1.02 + 0.747i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.36 - 0.374i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (2.60 + 5.11i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (0.610 - 1.88i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (3.96 - 0.628i)T + (21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-5.09 + 1.65i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (4.42 + 1.43i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.58 - 0.409i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (0.333 + 0.458i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-0.902 + 0.902i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.41 - 1.73i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-3.28 - 1.67i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-5.16 + 3.75i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.54 - 6.25i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-9.06 + 4.61i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (0.623 + 1.91i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.771 - 4.87i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (1.40 - 0.456i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (2.90 - 1.48i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (1.42 + 1.03i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.83 - 2.97i)T + (57.0 + 78.4i)T^{2} \) |
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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
−11.95180698714088173721697567775, −11.35577976039965510847861521145, −10.42341939019293917661412617268, −9.029793878607804277222097071274, −8.244936592140779214667656960160, −7.58123311828350452987685370095, −4.90961790485378547487611684680, −3.89005930147460502790763653262, −2.31539395419667636923329024324, −0.995361778458818864074151881249,
3.90189960048198991046128712291, 4.70281267475250318758024236937, 6.08057879866418759876715441973, 7.21406110814834129199214122358, 8.364276224635561611178722890301, 8.673903071920919011157925180977, 10.02999024250247955684569343865, 10.92109853548214307778948186101, 12.64648033241661589072055820131, 14.01229084074628629896385971993
