| L(s) = 1 | + (−0.387 − 1.44i)2-s + (3.29 − 5.70i)3-s + (11.9 − 6.87i)4-s + (−10.5 + 10.5i)5-s + (−9.52 − 2.55i)6-s + (−14.3 + 53.3i)7-s + (−31.4 − 31.4i)8-s + (18.8 + 32.5i)9-s + (19.3 + 11.1i)10-s + (−52.6 + 14.1i)11-s − 90.6i·12-s + (115. − 123. i)13-s + 82.7·14-s + (25.4 + 95.0i)15-s + (76.7 − 132. i)16-s + (−171. + 99.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.0968 − 0.361i)2-s + (0.365 − 0.633i)3-s + (0.744 − 0.429i)4-s + (−0.422 + 0.422i)5-s + (−0.264 − 0.0708i)6-s + (−0.291 + 1.08i)7-s + (−0.492 − 0.492i)8-s + (0.232 + 0.402i)9-s + (0.193 + 0.111i)10-s + (−0.435 + 0.116i)11-s − 0.629i·12-s + (0.684 − 0.729i)13-s + 0.421·14-s + (0.113 + 0.422i)15-s + (0.299 − 0.519i)16-s + (−0.593 + 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.712i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.702 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.12935 - 0.472428i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12935 - 0.472428i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-115. + 123. i)T \) |
| good | 2 | \( 1 + (0.387 + 1.44i)T + (-13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (-3.29 + 5.70i)T + (-40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (10.5 - 10.5i)T - 625iT^{2} \) |
| 7 | \( 1 + (14.3 - 53.3i)T + (-2.07e3 - 1.20e3i)T^{2} \) |
| 11 | \( 1 + (52.6 - 14.1i)T + (1.26e4 - 7.32e3i)T^{2} \) |
| 17 | \( 1 + (171. - 99.0i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (106. + 28.5i)T + (1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (761. + 439. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-398. + 690. i)T + (-3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-1.10e3 + 1.10e3i)T - 9.23e5iT^{2} \) |
| 37 | \( 1 + (1.20e3 - 322. i)T + (1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 + (-802. - 2.99e3i)T + (-2.44e6 + 1.41e6i)T^{2} \) |
| 43 | \( 1 + (798. - 461. i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (995. + 995. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 - 3.86e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-216. + 807. i)T + (-1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-1.38e3 - 2.39e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (783. + 2.92e3i)T + (-1.74e7 + 1.00e7i)T^{2} \) |
| 71 | \( 1 + (-4.08e3 - 1.09e3i)T + (2.20e7 + 1.27e7i)T^{2} \) |
| 73 | \( 1 + (918. + 918. i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 + 7.90e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (-2.97e3 + 2.97e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (-1.21e4 + 3.26e3i)T + (5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (7.62e3 + 2.04e3i)T + (7.66e7 + 4.42e7i)T^{2} \) |
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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
−19.02588351982128906408934075598, −18.28410556337792147455582376352, −15.89992406583644272335764744618, −15.07417218482899565828511795157, −13.14692736900172640435739909296, −11.77763062507697912819195651347, −10.30332323151845263401021695455, −8.110818512787731451654047049585, −6.28334529766259157779268461007, −2.46366264923141106715943862682,
3.88071576187802076261343566212, 6.86369860537749127240893058850, 8.559039382915840624199973215919, 10.45805030490539247627411922020, 12.12013767611598169614328069619, 13.94138561091911678216834884649, 15.76788827876822647948808156417, 16.16035099439651953865205196829, 17.69423442256621739076964638903, 19.76118869781080229605074818601
