| L(s) = 1 | + (2.65 − 2.65i)2-s + (15.7 + 6.51i)3-s + 17.8i·4-s + (13.8 − 33.4i)5-s + (59.1 − 24.4i)6-s + (−40.7 − 98.4i)7-s + (132. + 132. i)8-s + (33.2 + 33.2i)9-s + (−52.0 − 125. i)10-s + (−521. + 215. i)11-s + (−116. + 281. i)12-s − 117. i·13-s + (−369. − 153. i)14-s + (435. − 435. i)15-s + 133.·16-s + (−991. + 661. i)17-s + ⋯ |
| L(s) = 1 | + (0.469 − 0.469i)2-s + (1.00 + 0.418i)3-s + 0.558i·4-s + (0.247 − 0.598i)5-s + (0.670 − 0.277i)6-s + (−0.314 − 0.759i)7-s + (0.732 + 0.732i)8-s + (0.136 + 0.136i)9-s + (−0.164 − 0.397i)10-s + (−1.29 + 0.538i)11-s + (−0.233 + 0.563i)12-s − 0.192i·13-s + (−0.504 − 0.208i)14-s + (0.500 − 0.500i)15-s + 0.129·16-s + (−0.831 + 0.555i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.149i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.08117 - 0.156899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08117 - 0.156899i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (991. - 661. i)T \) |
| good | 2 | \( 1 + (-2.65 + 2.65i)T - 32iT^{2} \) |
| 3 | \( 1 + (-15.7 - 6.51i)T + (171. + 171. i)T^{2} \) |
| 5 | \( 1 + (-13.8 + 33.4i)T + (-2.20e3 - 2.20e3i)T^{2} \) |
| 7 | \( 1 + (40.7 + 98.4i)T + (-1.18e4 + 1.18e4i)T^{2} \) |
| 11 | \( 1 + (521. - 215. i)T + (1.13e5 - 1.13e5i)T^{2} \) |
| 13 | \( 1 + 117. iT - 3.71e5T^{2} \) |
| 19 | \( 1 + (-771. + 771. i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 + (-3.09e3 + 1.28e3i)T + (4.55e6 - 4.55e6i)T^{2} \) |
| 29 | \( 1 + (-46.9 + 113. i)T + (-1.45e7 - 1.45e7i)T^{2} \) |
| 31 | \( 1 + (4.15e3 + 1.72e3i)T + (2.02e7 + 2.02e7i)T^{2} \) |
| 37 | \( 1 + (-9.69e3 - 4.01e3i)T + (4.90e7 + 4.90e7i)T^{2} \) |
| 41 | \( 1 + (-30.9 - 74.7i)T + (-8.19e7 + 8.19e7i)T^{2} \) |
| 43 | \( 1 + (-1.57e4 - 1.57e4i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 - 1.19e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + (-3.01e3 + 3.01e3i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + (2.42e4 + 2.42e4i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (-1.03e4 - 2.48e4i)T + (-5.97e8 + 5.97e8i)T^{2} \) |
| 67 | \( 1 + 5.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (6.66e4 + 2.76e4i)T + (1.27e9 + 1.27e9i)T^{2} \) |
| 73 | \( 1 + (-2.71e4 + 6.55e4i)T + (-1.46e9 - 1.46e9i)T^{2} \) |
| 79 | \( 1 + (-4.54e4 + 1.88e4i)T + (2.17e9 - 2.17e9i)T^{2} \) |
| 83 | \( 1 + (2.50e4 - 2.50e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.06e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.45e4 + 1.55e5i)T + (-6.07e9 - 6.07e9i)T^{2} \) |
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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
−17.70625385907140211996775190193, −16.44559906107051299719134333209, −15.00566372293682972231268627980, −13.43274063883320359042415252159, −12.85758596778288997149712820443, −10.81723850290025033439921319431, −9.146213047900990774088295353179, −7.70660744230555438535222501477, −4.55620989376744072391557687222, −2.85200213474244288119834684756,
2.64426367107246817690060564807, 5.57498671167718129810697902802, 7.28821719226368689312022746058, 9.074634795175423576051992346184, 10.77558916314508082743317888220, 13.08442624252393596213451456382, 13.97069371288215318681227877545, 15.02682159573339320428957874418, 16.05819477597166455829053545178, 18.40608260973397290120366782247
