| L(s) = 1 | + (0.694 + 3.93i)2-s + (1.57 − 8.95i)3-s + (−15.0 + 5.47i)4-s + (22.3 + 18.7i)5-s + 36.3·6-s + (−145. − 122. i)7-s + (−32 − 55.4i)8-s + (150. + 54.8i)9-s + (−58.3 + 101. i)10-s + (−198. − 344. i)11-s + (25.2 + 143. i)12-s + (−33.1 + 12.0i)13-s + (379. − 657. i)14-s + (203. − 170. i)15-s + (196. − 164. i)16-s + (−1.65e3 − 603. i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.101 − 0.574i)3-s + (−0.469 + 0.171i)4-s + (0.399 + 0.335i)5-s + 0.412·6-s + (−1.12 − 0.941i)7-s + (−0.176 − 0.306i)8-s + (0.620 + 0.225i)9-s + (−0.184 + 0.319i)10-s + (−0.495 − 0.858i)11-s + (0.0506 + 0.287i)12-s + (−0.0544 + 0.0198i)13-s + (0.517 − 0.896i)14-s + (0.233 − 0.195i)15-s + (0.191 − 0.160i)16-s + (−1.39 − 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.874733 - 0.789627i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.874733 - 0.789627i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.694 - 3.93i)T \) |
| 37 | \( 1 + (4.63e3 + 6.91e3i)T \) |
| good | 3 | \( 1 + (-1.57 + 8.95i)T + (-228. - 83.1i)T^{2} \) |
| 5 | \( 1 + (-22.3 - 18.7i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (145. + 122. i)T + (2.91e3 + 1.65e4i)T^{2} \) |
| 11 | \( 1 + (198. + 344. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (33.1 - 12.0i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (1.65e3 + 603. i)T + (1.08e6 + 9.12e5i)T^{2} \) |
| 19 | \( 1 + (-490. + 2.78e3i)T + (-2.32e6 - 8.46e5i)T^{2} \) |
| 23 | \( 1 + (-641. + 1.11e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-4.44e3 - 7.69e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 - 2.90e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (3.83e3 - 1.39e3i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + 2.12e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.41e4 - 2.44e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (3.38e3 - 2.83e3i)T + (7.26e7 - 4.11e8i)T^{2} \) |
| 59 | \( 1 + (-1.29e4 + 1.08e4i)T + (1.24e8 - 7.04e8i)T^{2} \) |
| 61 | \( 1 + (-1.44e4 + 5.25e3i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-5.57e4 - 4.67e4i)T + (2.34e8 + 1.32e9i)T^{2} \) |
| 71 | \( 1 + (267. - 1.51e3i)T + (-1.69e9 - 6.17e8i)T^{2} \) |
| 73 | \( 1 - 3.63e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.18e4 + 1.83e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (4.97e4 + 1.80e4i)T + (3.01e9 + 2.53e9i)T^{2} \) |
| 89 | \( 1 + (-3.00e4 + 2.51e4i)T + (9.69e8 - 5.49e9i)T^{2} \) |
| 97 | \( 1 + (7.86e4 - 1.36e5i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−13.41463835457718174772620125955, −12.83773483366685499459318882630, −10.96588262796331423506074266934, −9.876101693468000735803013096461, −8.541761100846672557647332270281, −6.89783862496295162438904674108, −6.69996293618017162801078754875, −4.74113477668582043284430197718, −2.91415337563889234726457943309, −0.47727493872557657739182832336,
1.98307117024872153257395105769, 3.60143041097942603310259057740, 5.04137428774702578412882866706, 6.46893868191517727983141610118, 8.509881484023325095583562966022, 9.879795590164055573637749913795, 9.952324137674879826475791397586, 11.82485765733009665706107920127, 12.74285836127144484138704422551, 13.45838889737136408761485830148
