| L(s) = 1 | + (−1.68 + 7.37i)2-s + (−6.81 + 8.54i)3-s + (−22.7 − 10.9i)4-s + (−28.2 + 35.4i)5-s + (−51.5 − 64.6i)6-s + (−64.4 − 112. i)7-s + (−31.6 + 39.6i)8-s + (27.5 + 120. i)9-s + (−213. − 267. i)10-s + (143. − 628. i)11-s + (249. − 119. i)12-s + (−171. + 750. i)13-s + (938. − 286. i)14-s + (−110. − 482. i)15-s + (−744. − 933. i)16-s + (1.09e3 − 527. i)17-s + ⋯ |
| L(s) = 1 | + (−0.297 + 1.30i)2-s + (−0.436 + 0.547i)3-s + (−0.712 − 0.343i)4-s + (−0.505 + 0.633i)5-s + (−0.584 − 0.733i)6-s + (−0.497 − 0.867i)7-s + (−0.174 + 0.219i)8-s + (0.113 + 0.496i)9-s + (−0.675 − 0.847i)10-s + (0.357 − 1.56i)11-s + (0.499 − 0.240i)12-s + (−0.281 + 1.23i)13-s + (1.27 − 0.390i)14-s + (−0.126 − 0.553i)15-s + (−0.726 − 0.911i)16-s + (0.919 − 0.442i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.213656 - 0.179139i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.213656 - 0.179139i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (64.4 + 112. i)T \) |
| good | 2 | \( 1 + (1.68 - 7.37i)T + (-28.8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (6.81 - 8.54i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (28.2 - 35.4i)T + (-695. - 3.04e3i)T^{2} \) |
| 11 | \( 1 + (-143. + 628. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (171. - 750. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-1.09e3 + 527. i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 - 1.08e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (3.67e3 + 1.76e3i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 29 | \( 1 + (4.58e3 - 2.20e3i)T + (1.27e7 - 1.60e7i)T^{2} \) |
| 31 | \( 1 + 7.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (1.48e4 - 7.14e3i)T + (4.32e7 - 5.42e7i)T^{2} \) |
| 41 | \( 1 + (-8.60e3 + 1.07e4i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 43 | \( 1 + (-6.71e3 - 8.41e3i)T + (-3.27e7 + 1.43e8i)T^{2} \) |
| 47 | \( 1 + (-372. + 1.63e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (1.18e4 + 5.72e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-7.90e3 - 9.91e3i)T + (-1.59e8 + 6.96e8i)T^{2} \) |
| 61 | \( 1 + (-2.05e4 + 9.90e3i)T + (5.26e8 - 6.60e8i)T^{2} \) |
| 67 | \( 1 + 3.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (605. + 291. i)T + (1.12e9 + 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-6.59e3 - 2.88e4i)T + (-1.86e9 + 8.99e8i)T^{2} \) |
| 79 | \( 1 - 1.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-1.68e4 - 7.39e4i)T + (-3.54e9 + 1.70e9i)T^{2} \) |
| 89 | \( 1 + (-8.87e3 - 3.88e4i)T + (-5.03e9 + 2.42e9i)T^{2} \) |
| 97 | \( 1 - 7.68e4T + 8.58e9T^{2} \) |
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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
−16.08839361339401437201107120892, −14.41935936834892512920553803715, −13.87174291141227071973244117083, −11.69784366182471851660938453374, −10.74966733433769103353000218320, −9.323799365803888704364576468255, −7.77837051241166126252844904610, −6.83863391262224484118052592964, −5.52636642271911100213761839855, −3.69649127456067652265882312761,
0.16391949764134634543355015684, 1.76800875231296144824404258870, 3.65928476741038548951480900437, 5.74728124513097433036429062574, 7.52086307591900926427726548110, 9.253998164588256383496574840671, 10.08139994461941190746760764307, 11.76559004968280756547093998828, 12.43032215760862213073683387718, 12.71223934911615367561483202071
