L(s) = 1 | + (0.900 − 0.520i)2-s + (0.384 + 0.666i)3-s + (−0.459 + 0.795i)4-s + 1.67i·5-s + (0.692 + 0.400i)6-s + 3.03i·8-s + (1.20 − 2.08i)9-s + (0.871 + 1.51i)10-s + (−0.465 + 0.268i)11-s − 0.706·12-s + (−1.96 + 3.02i)13-s + (−1.11 + 0.644i)15-s + (0.660 + 1.14i)16-s + (−2.81 + 4.87i)17-s − 2.50i·18-s + (1.74 + 1.00i)19-s + ⋯ |
L(s) = 1 | + (0.636 − 0.367i)2-s + (0.222 + 0.384i)3-s + (−0.229 + 0.397i)4-s + 0.749i·5-s + (0.282 + 0.163i)6-s + 1.07i·8-s + (0.401 − 0.695i)9-s + (0.275 + 0.477i)10-s + (−0.140 + 0.0810i)11-s − 0.203·12-s + (−0.544 + 0.838i)13-s + (−0.288 + 0.166i)15-s + (0.165 + 0.285i)16-s + (−0.682 + 1.18i)17-s − 0.590i·18-s + (0.400 + 0.231i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0651 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0651 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36305 + 1.27699i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36305 + 1.27699i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (1.96 - 3.02i)T \) |
good | 2 | \( 1 + (-0.900 + 0.520i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.384 - 0.666i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 - 1.67iT - 5T^{2} \) |
| 11 | \( 1 + (0.465 - 0.268i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.81 - 4.87i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.74 - 1.00i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.33 + 5.78i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.43 - 4.22i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.78iT - 31T^{2} \) |
| 37 | \( 1 + (1.19 - 0.690i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.07 - 0.620i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.63 - 2.82i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.79iT - 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + (-6.99 - 4.03i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.88 + 3.27i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.31 + 4.79i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.31 - 0.760i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 15.6iT - 73T^{2} \) |
| 79 | \( 1 - 8.26T + 79T^{2} \) |
| 83 | \( 1 + 9.42iT - 83T^{2} \) |
| 89 | \( 1 + (1.85 - 1.06i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.58 + 3.22i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.73255717235082295434791760594, −10.14622856091441640279586811333, −8.995819223590451733982309149533, −8.356194337872060188506280407968, −7.08678949893146958879184790947, −6.37118697574095996967744547316, −4.92586735611503281992335283776, −4.09739990837099234966040230775, −3.29593141638208951485556941611, −2.17159687836918048572391607238,
0.835639290347553603545531251760, 2.45613471600715616653870847156, 4.03555722985183111525094227258, 5.05730916481408212626687613975, 5.48305314005079172422003706830, 6.83879512704145524664727892411, 7.57758550248101298073988930160, 8.543529240809447462618422407234, 9.591222955021486627353945437790, 10.15567528184694856283926876334