| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (0.222 − 0.0484i)3-s + (−0.989 − 0.142i)4-s + (−1.95 + 1.08i)5-s + (0.0324 + 0.225i)6-s + (−2.08 + 1.13i)7-s + (0.212 − 0.977i)8-s + (−2.68 + 1.22i)9-s + (−0.938 − 2.02i)10-s + (1.23 + 1.06i)11-s + (−0.227 + 0.0162i)12-s + (−1.52 + 2.79i)13-s + (−0.987 − 2.16i)14-s + (−0.383 + 0.335i)15-s + (0.959 + 0.281i)16-s + (0.715 + 0.535i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (0.128 − 0.0279i)3-s + (−0.494 − 0.0711i)4-s + (−0.875 + 0.483i)5-s + (0.0132 + 0.0920i)6-s + (−0.788 + 0.430i)7-s + (0.0751 − 0.345i)8-s + (−0.893 + 0.408i)9-s + (−0.296 − 0.641i)10-s + (0.372 + 0.322i)11-s + (−0.0655 + 0.00469i)12-s + (−0.423 + 0.776i)13-s + (−0.264 − 0.578i)14-s + (−0.0989 + 0.0866i)15-s + (0.239 + 0.0704i)16-s + (0.173 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0808163 + 0.618038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0808163 + 0.618038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 5 | \( 1 + (1.95 - 1.08i)T \) |
| 23 | \( 1 + (-0.0633 - 4.79i)T \) |
| good | 3 | \( 1 + (-0.222 + 0.0484i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (2.08 - 1.13i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-1.23 - 1.06i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.52 - 2.79i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-0.715 - 0.535i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.289 + 2.01i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-5.97 + 0.858i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (1.24 - 0.798i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (8.73 + 3.25i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-3.61 + 7.91i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.40 - 6.44i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-7.09 - 7.09i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.13 - 7.57i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-0.585 - 1.99i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.0788 - 0.122i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (1.90 + 0.136i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-5.63 - 6.50i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-3.44 - 4.60i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (3.63 - 1.06i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.217 - 0.583i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (13.2 + 8.49i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.18 + 5.86i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−12.54446980920989930844291399960, −11.80141427202209319605211974262, −10.74486618989714175097559058629, −9.458706640932435112506785686565, −8.683406209413896960240116525244, −7.54184762690723134204613082135, −6.74843550509183764726178090724, −5.59774124197015957737409878733, −4.17888231301420702813914008589, −2.83815048373017404851583973688,
0.50684901059920367048518778352, 3.00083797786618795383477382875, 3.88210615299853324781917289446, 5.28999208846816840765853530283, 6.73208853404746459704777608707, 8.114713915078813324433043788454, 8.812824032638987523117782191250, 9.926092822237008573431263397500, 10.83806337608685699346507393196, 11.97503226163321194701126925397
