| L(s) = 1 | + (0.0713 − 0.997i)2-s + (−3.06 + 0.666i)3-s + (−0.989 − 0.142i)4-s + (−1.99 + 1.01i)5-s + (0.446 + 3.10i)6-s + (2.74 − 1.50i)7-s + (−0.212 + 0.977i)8-s + (6.21 − 2.83i)9-s + (0.873 + 2.05i)10-s + (3.61 + 3.12i)11-s + (3.12 − 0.223i)12-s + (1.90 − 3.48i)13-s + (−1.30 − 2.84i)14-s + (5.42 − 4.44i)15-s + (0.959 + 0.281i)16-s + (−1.03 − 0.775i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (−1.76 + 0.384i)3-s + (−0.494 − 0.0711i)4-s + (−0.890 + 0.455i)5-s + (0.182 + 1.26i)6-s + (1.03 − 0.567i)7-s + (−0.0751 + 0.345i)8-s + (2.07 − 0.946i)9-s + (0.276 + 0.650i)10-s + (1.08 + 0.943i)11-s + (0.902 − 0.0645i)12-s + (0.528 − 0.967i)13-s + (−0.347 − 0.761i)14-s + (1.39 − 1.14i)15-s + (0.239 + 0.0704i)16-s + (−0.251 − 0.188i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.686255 - 0.0978059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.686255 - 0.0978059i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.99 - 1.01i)T \) |
| 23 | \( 1 + (-0.105 - 4.79i)T \) |
| good | 3 | \( 1 + (3.06 - 0.666i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-2.74 + 1.50i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-3.61 - 3.12i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.90 + 3.48i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (1.03 + 0.775i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.898 - 6.24i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-6.15 + 0.884i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.07 + 3.26i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-2.17 - 0.812i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-1.61 + 3.53i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.38 + 6.36i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-3.13 - 3.13i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.37 - 4.34i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-0.731 - 2.49i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (1.44 + 2.25i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (0.563 + 0.0402i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-3.48 - 4.01i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-3.19 - 4.27i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (2.54 - 0.748i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (2.59 - 6.96i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (3.18 + 2.04i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-0.929 - 2.49i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−11.94920569313583573246650234513, −11.24470032725082441004608004093, −10.58144634747244957749673036669, −9.854116735988807616470993837692, −8.111202413383128245892454147289, −7.01449978752963446089679449698, −5.79179670126465219963331848376, −4.52641361791728016881875029993, −3.92486521980359348573758636908, −1.14383812674494418881813258669,
0.994737132796726042927782944520, 4.37760157260916620974085950858, 4.95016549804863315543069089932, 6.27566220001743820743743115867, 6.82163290929736667607523498090, 8.239782014147550622624900487685, 8.966413584916360006922318126623, 10.81964029267200946213838603741, 11.55604589923423419760642960714, 11.90851884044266938093728115600
