| L(s) = 1 | + (1.20 − 0.747i)2-s + (−1.00 − 0.0575i)3-s + (0.00432 − 0.00872i)4-s + (1.65 + 1.50i)5-s + (−1.24 + 0.679i)6-s + (−3.19 + 1.81i)7-s + (0.269 + 2.80i)8-s + (−1.98 − 0.228i)9-s + (3.11 + 0.575i)10-s + (−3.81 + 3.67i)11-s + (−0.00482 + 0.00848i)12-s + (2.45 − 0.936i)13-s + (−2.48 + 4.57i)14-s + (−1.56 − 1.60i)15-s + (2.43 + 3.19i)16-s + (−1.80 − 3.47i)17-s + ⋯ |
| L(s) = 1 | + (0.851 − 0.528i)2-s + (−0.577 − 0.0332i)3-s + (0.00216 − 0.00436i)4-s + (0.739 + 0.672i)5-s + (−0.509 + 0.277i)6-s + (−1.20 + 0.686i)7-s + (0.0953 + 0.992i)8-s + (−0.660 − 0.0762i)9-s + (0.985 + 0.181i)10-s + (−1.15 + 1.10i)11-s + (−0.00139 + 0.00244i)12-s + (0.681 − 0.259i)13-s + (−0.664 + 1.22i)14-s + (−0.405 − 0.413i)15-s + (0.609 + 0.799i)16-s + (−0.437 − 0.843i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.950372 + 0.852369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.950372 + 0.852369i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.65 - 1.50i)T \) |
| 83 | \( 1 + (-4.97 + 7.62i)T \) |
| good | 2 | \( 1 + (-1.20 + 0.747i)T + (0.887 - 1.79i)T^{2} \) |
| 3 | \( 1 + (1.00 + 0.0575i)T + (2.98 + 0.344i)T^{2} \) |
| 7 | \( 1 + (3.19 - 1.81i)T + (3.57 - 6.01i)T^{2} \) |
| 11 | \( 1 + (3.81 - 3.67i)T + (0.421 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.45 + 0.936i)T + (9.70 - 8.64i)T^{2} \) |
| 17 | \( 1 + (1.80 + 3.47i)T + (-9.78 + 13.9i)T^{2} \) |
| 19 | \( 1 + (0.407 + 2.09i)T + (-17.6 + 7.10i)T^{2} \) |
| 23 | \( 1 + (-4.37 - 2.06i)T + (14.6 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-1.01 - 2.51i)T + (-20.8 + 20.1i)T^{2} \) |
| 31 | \( 1 + (-4.70 + 3.87i)T + (5.90 - 30.4i)T^{2} \) |
| 37 | \( 1 + (-2.75 + 3.47i)T + (-8.43 - 36.0i)T^{2} \) |
| 41 | \( 1 + (-7.21 + 1.68i)T + (36.7 - 18.1i)T^{2} \) |
| 43 | \( 1 + (-2.15 - 12.3i)T + (-40.4 + 14.5i)T^{2} \) |
| 47 | \( 1 + (-1.36 - 6.37i)T + (-42.8 + 19.2i)T^{2} \) |
| 53 | \( 1 + (-1.17 + 5.49i)T + (-48.3 - 21.6i)T^{2} \) |
| 59 | \( 1 + (9.85 - 0.377i)T + (58.8 - 4.51i)T^{2} \) |
| 61 | \( 1 + (2.84 - 1.69i)T + (29.1 - 53.5i)T^{2} \) |
| 67 | \( 1 + (0.906 + 0.122i)T + (64.6 + 17.7i)T^{2} \) |
| 71 | \( 1 + (3.79 - 1.04i)T + (61.0 - 36.2i)T^{2} \) |
| 73 | \( 1 + (-9.17 - 12.5i)T + (-22.0 + 69.5i)T^{2} \) |
| 79 | \( 1 + (14.9 + 7.40i)T + (47.8 + 62.8i)T^{2} \) |
| 89 | \( 1 + (-2.57 - 4.73i)T + (-48.3 + 74.7i)T^{2} \) |
| 97 | \( 1 + (1.56 + 5.30i)T + (-81.4 + 52.7i)T^{2} \) |
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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
−11.38057631772790605201275883219, −10.88194986426425734409888491938, −9.757589916587723082917822393834, −8.957323047114914435447059989250, −7.52248354525031544366637768798, −6.33032285809444206189863622324, −5.63567104222652118374373800645, −4.70772921878517171071336062403, −2.93735301261891075282609733123, −2.63894739969197009904966370489,
0.65383511038694819900895285998, 3.07515790490705767119005781690, 4.34184687748445742118010335191, 5.47481175050100715944745191456, 6.05103450607734166344457431666, 6.65935508885478528182248474841, 8.265623742081658731811619770697, 9.188349685023818087609663666573, 10.39510233018915619411693748120, 10.71631635897584496213694211767
