L(s) = 1 | + (1.35 − 2.34i)5-s + (−10.0 + 5.79i)7-s + (8.54 − 4.93i)11-s + (−0.296 + 0.513i)13-s + 8.87·17-s − 14.0i·19-s + (−18.2 − 10.5i)23-s + (8.82 + 15.2i)25-s + (10.1 + 17.6i)29-s + (−14.3 − 8.27i)31-s + 31.4i·35-s + 40.6·37-s + (−21.2 + 36.7i)41-s + (−32.2 + 18.6i)43-s + (1.57 − 0.907i)47-s + ⋯ |
L(s) = 1 | + (0.271 − 0.469i)5-s + (−1.43 + 0.828i)7-s + (0.777 − 0.448i)11-s + (−0.0227 + 0.0394i)13-s + 0.522·17-s − 0.742i·19-s + (−0.794 − 0.458i)23-s + (0.352 + 0.611i)25-s + (0.350 + 0.607i)29-s + (−0.462 − 0.266i)31-s + 0.898i·35-s + 1.09·37-s + (−0.517 + 0.896i)41-s + (−0.749 + 0.432i)43-s + (0.0334 − 0.0193i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.09518763182\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09518763182\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.35 + 2.34i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (10.0 - 5.79i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.54 + 4.93i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (0.296 - 0.513i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 8.87T + 289T^{2} \) |
| 19 | \( 1 + 14.0iT - 361T^{2} \) |
| 23 | \( 1 + (18.2 + 10.5i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-10.1 - 17.6i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (14.3 + 8.27i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 40.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (21.2 - 36.7i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (32.2 - 18.6i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-1.57 + 0.907i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 21.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (76.6 + 44.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (36.4 + 63.2i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (38.3 + 22.1i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 111. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 76.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-8.30 + 4.79i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (73.6 - 42.5i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 64.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + (3.59 + 6.22i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.446315702899143527033566552578, −8.911434038953980524969204543600, −8.098695010862454809550731584268, −6.89447466923757956039013853176, −6.27107297365908815426767405114, −5.62159267813210178924157379516, −4.60568498904405027901513722616, −3.44337193113416516718272833888, −2.74748174934366272634054127377, −1.36587670546414484891444569351,
0.02550347047533722113401493656, 1.45224044012686706142442617258, 2.81294020756001872013630986524, 3.66038376564673602673034643740, 4.36892270743691681503039521458, 5.81639869431152901159098151489, 6.36466336732337851931003403137, 7.09359512654840248943784516930, 7.79040070881351270361541564874, 8.939832940371508241811909217171