L(s) = 1 | + i·2-s − 0.406i·3-s − 4-s + 0.406·6-s + 2.91i·7-s − i·8-s + 2.83·9-s + 6.51·11-s + 0.406i·12-s + 0.813i·13-s − 2.91·14-s + 16-s − 2.51i·17-s + 2.83i·18-s − 0.406·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.234i·3-s − 0.5·4-s + 0.166·6-s + 1.10i·7-s − 0.353i·8-s + 0.944·9-s + 1.96·11-s + 0.117i·12-s + 0.225i·13-s − 0.779·14-s + 0.250·16-s − 0.608i·17-s + 0.668i·18-s − 0.0933·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.024357933\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.024357933\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
good | 3 | \( 1 + 0.406iT - 3T^{2} \) |
| 7 | \( 1 - 2.91iT - 7T^{2} \) |
| 11 | \( 1 - 6.51T + 11T^{2} \) |
| 13 | \( 1 - 0.813iT - 13T^{2} \) |
| 17 | \( 1 + 2.51iT - 17T^{2} \) |
| 19 | \( 1 + 0.406T + 19T^{2} \) |
| 23 | \( 1 + 5.02iT - 23T^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 + 8.75T + 31T^{2} \) |
| 41 | \( 1 - 6.34T + 41T^{2} \) |
| 43 | \( 1 - 7.32iT - 43T^{2} \) |
| 47 | \( 1 + 5.42iT - 47T^{2} \) |
| 53 | \( 1 - 2.34iT - 53T^{2} \) |
| 59 | \( 1 - 1.42T + 59T^{2} \) |
| 61 | \( 1 + 1.32T + 61T^{2} \) |
| 67 | \( 1 + 5.42iT - 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 11.0iT - 73T^{2} \) |
| 79 | \( 1 - 1.75T + 79T^{2} \) |
| 83 | \( 1 - 7.05iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 2.34iT - 97T^{2} \) |
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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
−9.228470205736900585774089090771, −8.693889286351439285529710443431, −7.74385933721146117816726907187, −6.72473783347739364742459170913, −6.51522630809358657190874370344, −5.47659763544081397398694810382, −4.50215421854489219243444353815, −3.77151722058445054946407084986, −2.34386161067995139994168502255, −1.13456629758548915677350851613,
1.00958751302124253563761879127, 1.77263646420677980862884008897, 3.51806011294757786392347036976, 3.93040239074190205832488305015, 4.62776441898860035357898879527, 5.87813113095379794617133851337, 6.87792052793118570192405516541, 7.42562264000474507715618978506, 8.528273703360519890398857964619, 9.401227086439273409389217932574