| L(s) = 1 | + (−0.453 − 0.891i)2-s + (−0.587 + 0.809i)4-s + (−0.194 − 1.22i)5-s + (−2.03 + 1.73i)7-s + (0.987 + 0.156i)8-s + (−1.00 + 0.731i)10-s + (2.00 + 1.22i)11-s + (−0.277 + 0.0218i)13-s + (2.46 + 1.02i)14-s + (−0.309 − 0.951i)16-s + (6.85 − 1.64i)17-s + (0.483 + 0.0380i)19-s + (1.10 + 0.564i)20-s + (0.184 − 2.34i)22-s + (1.61 − 4.97i)23-s + ⋯ |
| L(s) = 1 | + (−0.321 − 0.630i)2-s + (−0.293 + 0.404i)4-s + (−0.0870 − 0.549i)5-s + (−0.768 + 0.656i)7-s + (0.349 + 0.0553i)8-s + (−0.318 + 0.231i)10-s + (0.604 + 0.370i)11-s + (−0.0769 + 0.00605i)13-s + (0.660 + 0.273i)14-s + (−0.0772 − 0.237i)16-s + (1.66 − 0.398i)17-s + (0.111 + 0.00873i)19-s + (0.247 + 0.126i)20-s + (0.0393 − 0.499i)22-s + (0.336 − 1.03i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03753 - 0.575973i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03753 - 0.575973i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.453 + 0.891i)T \) |
| 3 | \( 1 \) |
| 41 | \( 1 + (-5.48 - 3.30i)T \) |
| good | 5 | \( 1 + (0.194 + 1.22i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (2.03 - 1.73i)T + (1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (-2.00 - 1.22i)T + (4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (0.277 - 0.0218i)T + (12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (-6.85 + 1.64i)T + (15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (-0.483 - 0.0380i)T + (18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (-1.61 + 4.97i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (0.372 + 0.0895i)T + (25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (3.76 + 5.18i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-5.65 - 4.10i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (1.38 - 0.706i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-3.32 + 3.89i)T + (-7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (-2.32 + 9.67i)T + (-47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (0.0292 + 0.00951i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.40 + 6.68i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (3.88 - 2.38i)T + (30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (-4.04 + 6.59i)T + (-32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (-9.66 - 9.66i)T + 73iT^{2} \) |
| 79 | \( 1 + (6.22 - 2.58i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 8.98iT - 83T^{2} \) |
| 89 | \( 1 + (5.24 + 6.14i)T + (-13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (2.34 + 3.83i)T + (-44.0 + 86.4i)T^{2} \) |
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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
−9.949672638524476877680969510788, −9.558837196600435728373063374193, −8.721744519742793127011867671096, −7.86909632267548235480190022973, −6.77800997840424899260206065745, −5.70670975366646507795718198512, −4.66578516583502451939004199622, −3.51434860137682543491797726017, −2.47518499663195365473994458847, −0.907759915308568976520229549422,
1.10534018812433700933135950251, 3.14101731133879073064333207879, 3.93936874080934673194772014725, 5.42370376611250174154544954359, 6.20616087498108121752962995710, 7.18975380273052516859506714801, 7.62111512820700567617795174740, 8.875746563186727091463837022062, 9.599296461349892337942308709396, 10.39947029371845150608497164183
