| L(s) = 1 | + (−2.46 + 7.59i)2-s + (−14.2 − 10.3i)3-s + (−25.7 − 18.7i)4-s + (27.5 − 48.6i)5-s + (113. − 82.7i)6-s + 40.0·7-s + (−0.963 + 0.699i)8-s + (20.7 + 63.9i)9-s + (301. + 329. i)10-s + (184. − 566. i)11-s + (173. + 533. i)12-s + (−219. − 676. i)13-s + (−98.9 + 304. i)14-s + (−896. + 407. i)15-s + (−317. − 978. i)16-s + (−1.06e3 + 775. i)17-s + ⋯ |
| L(s) = 1 | + (−0.436 + 1.34i)2-s + (−0.914 − 0.664i)3-s + (−0.805 − 0.585i)4-s + (0.493 − 0.869i)5-s + (1.29 − 0.938i)6-s + 0.309·7-s + (−0.00532 + 0.00386i)8-s + (0.0854 + 0.263i)9-s + (0.952 + 1.04i)10-s + (0.459 − 1.41i)11-s + (0.347 + 1.06i)12-s + (−0.360 − 1.11i)13-s + (−0.134 + 0.415i)14-s + (−1.02 + 0.467i)15-s + (−0.310 − 0.955i)16-s + (−0.896 + 0.651i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.704i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.710 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.661030 - 0.272221i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.661030 - 0.272221i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-27.5 + 48.6i)T \) |
| good | 2 | \( 1 + (2.46 - 7.59i)T + (-25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (14.2 + 10.3i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 40.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-184. + 566. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (219. + 676. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.06e3 - 775. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-600. + 436. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (662. - 2.04e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-530. - 385. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-2.83e3 + 2.06e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (4.25e3 + 1.30e4i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-3.58e3 - 1.10e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.77e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.62e3 - 1.91e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.58e3 - 1.14e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.07e4 + 3.30e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.39e4 - 4.28e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-3.43e4 + 2.49e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-3.01e4 - 2.19e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-2.02e4 + 6.21e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-6.43e4 - 4.67e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (6.00e4 - 4.36e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-4.58e3 + 1.41e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (8.05e4 + 5.84e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−16.66100572420763429124145360958, −15.56584176163403848850245657574, −13.98898855189674784396241187609, −12.66806543572318828893262528278, −11.26875584585758937241432925187, −9.129322443818032400667713802730, −7.914400811026178190311496602872, −6.25914041520000669775861139277, −5.45778868693507068337965697855, −0.62417039537777512042083922047,
2.15660546543804341221852346185, 4.49513135911680051686144764302, 6.68263977086984287014851394925, 9.434619266505316972454245562481, 10.28808316395866109794278965283, 11.30364324838907484217648515922, 12.16351183407502151668917087326, 14.07259381971252067922832098811, 15.53755881989227066051799705132, 17.19610687919497893358102064894
