L(s) = 1 | + 11.8·2-s + 59.5i·3-s − 371.·4-s − 633. i·5-s + 706. i·6-s − 1.09e4i·7-s − 1.04e4·8-s + 1.61e4·9-s − 7.51e3i·10-s − 2.61e4i·11-s − 2.21e4i·12-s − 1.32e5·13-s − 1.29e5i·14-s + 3.77e4·15-s + 6.59e4·16-s + (9.71e4 − 3.30e5i)17-s + ⋯ |
L(s) = 1 | + 0.524·2-s + 0.424i·3-s − 0.725·4-s − 0.453i·5-s + 0.222i·6-s − 1.72i·7-s − 0.904·8-s + 0.819·9-s − 0.237i·10-s − 0.539i·11-s − 0.308i·12-s − 1.28·13-s − 0.901i·14-s + 0.192·15-s + 0.251·16-s + (0.282 − 0.959i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.756031 - 1.01027i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.756031 - 1.01027i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-9.71e4 + 3.30e5i)T \) |
good | 2 | \( 1 - 11.8T + 512T^{2} \) |
| 3 | \( 1 - 59.5iT - 1.96e4T^{2} \) |
| 5 | \( 1 + 633. iT - 1.95e6T^{2} \) |
| 7 | \( 1 + 1.09e4iT - 4.03e7T^{2} \) |
| 11 | \( 1 + 2.61e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + 1.32e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 8.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.27e6iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 6.34e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + 8.52e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 + 8.05e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 2.36e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 - 2.89e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.63e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.45e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.84e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.18e7iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 1.46e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.08e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + 7.44e7iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 1.47e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + 5.69e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.03e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.03e9iT - 7.60e17T^{2} \) |
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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
−16.46081526179338120424383821961, −14.75202965942492090447556824266, −13.62114682408936828709348347325, −12.60825713534052527642986849032, −10.55439008114787342679777528113, −9.313295266902893474655960318994, −7.32504024181043796152315935774, −4.91697991010355192109924487251, −3.87529523327844400879732534899, −0.54305372379285256143183400678,
2.40934031802655787283714915957, 4.68981573459531458337087955075, 6.37585127962757766144936945760, 8.445182933151621519665809281087, 9.955773684703541598658957072006, 12.29640301331616995717342609577, 12.73338729580187310061686912228, 14.64014097140272102378393747829, 15.24213376538413085229801626561, 17.40764716051361783465724455481