L(s) = 1 | + (1.28 − 2.70i)3-s + (9.36 + 5.40i)7-s + (−5.67 − 6.98i)9-s + (7.78 + 4.49i)11-s + (3.96 − 2.28i)13-s + 7.55·17-s − 8.73·19-s + (26.7 − 18.3i)21-s + (4.02 + 6.96i)23-s + (−26.2 + 6.36i)27-s + (38.2 + 22.0i)29-s + (4.53 + 7.85i)31-s + (22.2 − 15.2i)33-s + 56.1i·37-s + (−1.08 − 13.6i)39-s + ⋯ |
L(s) = 1 | + (0.429 − 0.902i)3-s + (1.33 + 0.772i)7-s + (−0.630 − 0.776i)9-s + (0.708 + 0.408i)11-s + (0.304 − 0.175i)13-s + 0.444·17-s − 0.459·19-s + (1.27 − 0.875i)21-s + (0.174 + 0.302i)23-s + (−0.971 + 0.235i)27-s + (1.31 + 0.760i)29-s + (0.146 + 0.253i)31-s + (0.673 − 0.463i)33-s + 1.51i·37-s + (−0.0278 − 0.350i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.740489544\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.740489544\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.28 + 2.70i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-9.36 - 5.40i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.78 - 4.49i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-3.96 + 2.28i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 7.55T + 289T^{2} \) |
| 19 | \( 1 + 8.73T + 361T^{2} \) |
| 23 | \( 1 + (-4.02 - 6.96i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-38.2 - 22.0i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-4.53 - 7.85i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 56.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-53.9 + 31.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (34.7 + 20.0i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-9.10 + 15.7i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 23.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-59.3 + 34.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-16.3 + 28.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (7.20 - 4.15i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 115. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 125. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (15.5 - 26.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-77.0 + 133. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (143. + 83.1i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.686524787635577236238410588569, −8.593527185830431043624190982792, −8.396334861898356491998364794448, −7.35812838143141379108903583611, −6.51919944883766214407708623466, −5.54716629972293128318958357716, −4.56149868169430798639120436116, −3.21154921005753119054438838774, −2.04041607588994394061908770865, −1.18204800971794276871323068213,
1.09791085325561335622904007104, 2.52299861641441539596993980176, 3.91099198632266561530486719453, 4.39527327516247360781347877764, 5.39591379957862517917600441985, 6.51601977808206907190342922477, 7.77305196918889106519147764477, 8.280797830539341268802572410075, 9.128547476327079073328406139921, 10.00005305132951639908516024623