| L(s) = 1 | + 5.61i·5-s + 10.8·7-s − 9·9-s − 17.3i·11-s − 33.9·17-s + 19i·19-s − 34.8·23-s − 6.52·25-s + 61.0i·35-s + 31.1i·43-s − 50.5i·45-s − 93.2·47-s + 69.1·49-s + 97.5·55-s − 56.5i·61-s + ⋯ |
| L(s) = 1 | + 1.12i·5-s + 1.55·7-s − 9-s − 1.57i·11-s − 1.99·17-s + i·19-s − 1.51·23-s − 0.261·25-s + 1.74i·35-s + 0.725i·43-s − 1.12i·45-s − 1.98·47-s + 1.41·49-s + 1.77·55-s − 0.926i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1278575006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1278575006\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19iT \) |
| good | 3 | \( 1 + 9T^{2} \) |
| 5 | \( 1 - 5.61iT - 25T^{2} \) |
| 7 | \( 1 - 10.8T + 49T^{2} \) |
| 11 | \( 1 + 17.3iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 33.9T + 289T^{2} \) |
| 23 | \( 1 + 34.8T + 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 31.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 93.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 + 56.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 90iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−10.23300895124049356983899762391, −8.894181857281825228230947893227, −8.302930947503647951118948780355, −7.76183268534216701531056474690, −6.45129214748881852343240480497, −5.98706162934756056812717926667, −4.91045350113143179610580642923, −3.80580953289810200051144127071, −2.79065224034652483393403183579, −1.80154585676754745759645163154,
0.03412254195990647798593440929, 1.65860974424051416301467395533, 2.36653901302181592267897811152, 4.33869414273782042323153776602, 4.66232729717748279975963260361, 5.43450551088139933156804422115, 6.65230975253068370910404733304, 7.61976914507091920873822692600, 8.500981441174634441995889937894, 8.821231807962642380452529181313
