L(s) = 1 | + (−1.29 + 0.569i)2-s + (−0.908 + 1.47i)3-s + (1.35 − 1.47i)4-s + (0.335 − 2.42i)6-s − 2.50i·7-s + (−0.908 + 2.67i)8-s + (−1.35 − 2.67i)9-s − 3.36·11-s + (0.948 + 3.33i)12-s − 3.70·13-s + (1.42 + 3.24i)14-s + (−0.350 − 3.98i)16-s − 7.63i·17-s + (3.27 + 2.69i)18-s + 0.440i·19-s + ⋯ |
L(s) = 1 | + (−0.915 + 0.402i)2-s + (−0.524 + 0.851i)3-s + (0.675 − 0.737i)4-s + (0.136 − 0.990i)6-s − 0.948i·7-s + (−0.321 + 0.947i)8-s + (−0.450 − 0.892i)9-s − 1.01·11-s + (0.273 + 0.961i)12-s − 1.02·13-s + (0.382 + 0.868i)14-s + (−0.0876 − 0.996i)16-s − 1.85i·17-s + (0.771 + 0.635i)18-s + 0.100i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.295555 - 0.223138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.295555 - 0.223138i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.29 - 0.569i)T \) |
| 3 | \( 1 + (0.908 - 1.47i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2.50iT - 7T^{2} \) |
| 11 | \( 1 + 3.36T + 11T^{2} \) |
| 13 | \( 1 + 3.70T + 13T^{2} \) |
| 17 | \( 1 + 7.63iT - 17T^{2} \) |
| 19 | \( 1 - 0.440iT - 19T^{2} \) |
| 23 | \( 1 - 5.17T + 23T^{2} \) |
| 29 | \( 1 + 2.27iT - 29T^{2} \) |
| 31 | \( 1 + 3.39iT - 31T^{2} \) |
| 37 | \( 1 + 7.40T + 37T^{2} \) |
| 41 | \( 1 - 3.07iT - 41T^{2} \) |
| 43 | \( 1 + 8.40iT - 43T^{2} \) |
| 47 | \( 1 + 3.63T + 47T^{2} \) |
| 53 | \( 1 - 2.27iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 - 5.45iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 1.29T + 73T^{2} \) |
| 79 | \( 1 - 5.01iT - 79T^{2} \) |
| 83 | \( 1 + 1.81T + 83T^{2} \) |
| 89 | \( 1 + 5.35iT - 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−11.21967378310254057984271517228, −10.38174088947080060437316144887, −9.814974282656205383428285354516, −8.918744315837576878600919356855, −7.55054855015092814883072220977, −6.94094057728062594579758191906, −5.46504811515728168797790004881, −4.71501548691725087858585452637, −2.85796468558204773986180809540, −0.37040779619600545298043630020,
1.79824033868305525880523082077, 2.92526794406119207142279040446, 5.10488937975200745086128154332, 6.23290653187111138871373086109, 7.28481521658842475970704406953, 8.168916120863393892274265373458, 8.950659692556381359769357613694, 10.29458169522733644265770748146, 10.90035844669741893559451265413, 11.99310104648245425542062680758