| L(s) = 1 | + (−0.625 + 1.92i)2-s + 8.16i·3-s + (3.15 + 2.29i)4-s + (−13.3 − 9.67i)5-s + (−15.7 − 5.10i)6-s + (17.8 − 5.78i)7-s + (−19.4 + 14.1i)8-s − 39.6·9-s + (26.9 − 19.5i)10-s + (34.2 + 47.1i)11-s + (−18.7 + 25.8i)12-s + (−5.25 − 1.70i)13-s + 37.8i·14-s + (78.9 − 108. i)15-s + (−5.40 − 16.6i)16-s + (−3.83 − 5.28i)17-s + ⋯ |
| L(s) = 1 | + (−0.221 + 0.680i)2-s + 1.57i·3-s + (0.394 + 0.286i)4-s + (−1.19 − 0.865i)5-s + (−1.06 − 0.347i)6-s + (0.961 − 0.312i)7-s + (−0.861 + 0.625i)8-s − 1.46·9-s + (0.851 − 0.618i)10-s + (0.939 + 1.29i)11-s + (−0.451 + 0.620i)12-s + (−0.112 − 0.0364i)13-s + 0.723i·14-s + (1.35 − 1.87i)15-s + (−0.0844 − 0.259i)16-s + (−0.0547 − 0.0753i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.352561 + 1.08446i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.352561 + 1.08446i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 + (190. + 180. i)T \) |
| good | 2 | \( 1 + (0.625 - 1.92i)T + (-6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 - 8.16iT - 27T^{2} \) |
| 5 | \( 1 + (13.3 + 9.67i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (-17.8 + 5.78i)T + (277. - 201. i)T^{2} \) |
| 11 | \( 1 + (-34.2 - 47.1i)T + (-411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (5.25 + 1.70i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (3.83 + 5.28i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-102. + 33.3i)T + (5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (27.7 - 85.5i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-179. + 246. i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (7.03 - 5.11i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-70.1 - 50.9i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 43 | \( 1 + (-78.2 + 240. i)T + (-6.43e4 - 4.67e4i)T^{2} \) |
| 47 | \( 1 + (-81.0 - 26.3i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-221. + 304. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-106. + 328. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (196. + 605. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (532. - 733. i)T + (-9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-380. - 524. i)T + (-1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 - 260.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 181. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-245. + 79.6i)T + (5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-110. + 152. i)T + (-2.82e5 - 8.68e5i)T^{2} \) |
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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
−15.72834883753051112637317442566, −15.51471604499575773785077848747, −14.39150770479507039437576638620, −11.97632045117705210510856997283, −11.45554758030621324242361505652, −9.709033356468121293078618659892, −8.496712477300323850091880712691, −7.39365909383064589002982247766, −5.00066138580472897168973953158, −3.97490671628276076040271338767,
1.16438843441851064491571874650, 3.05584500124705203240108902197, 6.24847784126693824074810394438, 7.36203994389139092572885849214, 8.539626155662995147275866116468, 10.85698301127818191246337623927, 11.69676551093097427659392618864, 12.15285155548452003466633055500, 14.02622597453206318267084529859, 14.81997127979355402097732146080
