| L(s) = 1 | + (1.38 + 0.304i)2-s + (−0.677 + 1.93i)3-s + (1.81 + 0.840i)4-s + (−0.936 − 0.213i)5-s + (−1.52 + 2.46i)6-s + (−1.14 − 2.38i)7-s + (2.25 + 1.71i)8-s + (−0.948 − 0.756i)9-s + (−1.22 − 0.580i)10-s + (−1.04 − 0.117i)11-s + (−2.85 + 2.94i)12-s + (3.43 − 2.73i)13-s + (−0.859 − 3.64i)14-s + (1.04 − 1.67i)15-s + (2.58 + 3.05i)16-s + (−2.30 − 2.30i)17-s + ⋯ |
| L(s) = 1 | + (0.976 + 0.215i)2-s + (−0.391 + 1.11i)3-s + (0.907 + 0.420i)4-s + (−0.419 − 0.0956i)5-s + (−0.622 + 1.00i)6-s + (−0.433 − 0.901i)7-s + (0.795 + 0.605i)8-s + (−0.316 − 0.252i)9-s + (−0.388 − 0.183i)10-s + (−0.313 − 0.0353i)11-s + (−0.825 + 0.850i)12-s + (0.951 − 0.758i)13-s + (−0.229 − 0.973i)14-s + (0.270 − 0.431i)15-s + (0.646 + 0.762i)16-s + (−0.559 − 0.559i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29588 + 0.752749i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29588 + 0.752749i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 - 0.304i)T \) |
| 29 | \( 1 + (1.11 - 5.26i)T \) |
| good | 3 | \( 1 + (0.677 - 1.93i)T + (-2.34 - 1.87i)T^{2} \) |
| 5 | \( 1 + (0.936 + 0.213i)T + (4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (1.14 + 2.38i)T + (-4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (1.04 + 0.117i)T + (10.7 + 2.44i)T^{2} \) |
| 13 | \( 1 + (-3.43 + 2.73i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (2.30 + 2.30i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.0184 + 0.00646i)T + (14.8 - 11.8i)T^{2} \) |
| 23 | \( 1 + (-2.77 + 0.633i)T + (20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (5.46 + 8.70i)T + (-13.4 + 27.9i)T^{2} \) |
| 37 | \( 1 + (-1.09 - 9.74i)T + (-36.0 + 8.23i)T^{2} \) |
| 41 | \( 1 + (7.23 - 7.23i)T - 41iT^{2} \) |
| 43 | \( 1 + (-3.00 - 1.88i)T + (18.6 + 38.7i)T^{2} \) |
| 47 | \( 1 + (-0.163 + 1.45i)T + (-45.8 - 10.4i)T^{2} \) |
| 53 | \( 1 + (-2.32 + 10.1i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 - 9.95iT - 59T^{2} \) |
| 61 | \( 1 + (2.25 + 0.788i)T + (47.6 + 38.0i)T^{2} \) |
| 67 | \( 1 + (-6.72 + 8.43i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (-8.00 - 10.0i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.169 - 0.106i)T + (31.6 + 65.7i)T^{2} \) |
| 79 | \( 1 + (-0.148 - 1.31i)T + (-77.0 + 17.5i)T^{2} \) |
| 83 | \( 1 + (4.39 - 9.12i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (6.71 - 4.21i)T + (38.6 - 80.1i)T^{2} \) |
| 97 | \( 1 + (-11.3 + 3.98i)T + (75.8 - 60.4i)T^{2} \) |
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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
−13.55216455757513069346248300739, −13.00256288530776665680096935659, −11.46941067003744556136860897978, −10.83933823396053295402926848399, −9.857908209301332339091975375045, −8.131792819134230595772726275114, −6.86939194147524560017288850756, −5.48361217077257794250379178591, −4.35520357640368273431993351454, −3.41139566712933814359303408175,
1.99132678236549007981106801407, 3.78560387260806199070969226469, 5.60512838252621282345316087703, 6.48913595209447475036101861684, 7.47087481149933095497121485538, 9.038664035698351209893488441209, 10.78718997162283318438157025325, 11.67484640266100939399655353709, 12.46977470398928121636340296916, 13.13322531523388801866771207449
