L(s) = 1 | − i·2-s − 4-s + 1.79·5-s + (2.28 + 1.34i)7-s + i·8-s − 1.79i·10-s + 2.40i·11-s − 4.89i·13-s + (1.34 − 2.28i)14-s + 16-s + 3.66·17-s + 3.01i·19-s − 1.79·20-s + 2.40·22-s + 3.76i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.800·5-s + (0.861 + 0.506i)7-s + 0.353i·8-s − 0.566i·10-s + 0.724i·11-s − 1.35i·13-s + (0.358 − 0.609i)14-s + 0.250·16-s + 0.888·17-s + 0.692i·19-s − 0.400·20-s + 0.512·22-s + 0.785i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.006151952\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.006151952\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.28 - 1.34i)T \) |
good | 5 | \( 1 - 1.79T + 5T^{2} \) |
| 11 | \( 1 - 2.40iT - 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 - 3.66T + 17T^{2} \) |
| 19 | \( 1 - 3.01iT - 19T^{2} \) |
| 23 | \( 1 - 3.76iT - 23T^{2} \) |
| 29 | \( 1 - 6.56iT - 29T^{2} \) |
| 31 | \( 1 + 4.64iT - 31T^{2} \) |
| 37 | \( 1 - 9.36T + 37T^{2} \) |
| 41 | \( 1 - 8.08T + 41T^{2} \) |
| 43 | \( 1 - 6.96T + 43T^{2} \) |
| 47 | \( 1 + 5.13T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 11.3iT - 61T^{2} \) |
| 67 | \( 1 - 0.570T + 67T^{2} \) |
| 71 | \( 1 - 5.96iT - 71T^{2} \) |
| 73 | \( 1 + 12.3iT - 73T^{2} \) |
| 79 | \( 1 - 3.03T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 + 5.51iT - 97T^{2} \) |
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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
−9.690084028767756091707046894832, −9.301682600248394272659073470140, −7.967084640590606008972943074540, −7.69606835644865018870453876458, −5.98811925627827808347879918853, −5.49655181960804080757883530834, −4.57445914281695379593397597274, −3.30793730746271915850787244892, −2.25417618561486847018907915940, −1.28604922823270894401951735118,
1.10611197727195030109843867177, 2.47618831989350056195962651717, 4.04131041960535421513561566185, 4.76282771800891582947759938808, 5.82452671615489855782323529015, 6.42453936957718447046618170758, 7.44766737309465782842521769961, 8.129938399710131029676376671349, 9.083084625027708500773637051410, 9.648576849316772603455503509386