| L(s) = 1 | − 2.23i·5-s + 11.5i·7-s + 6.61i·11-s − 4.35·13-s + 15.5i·17-s − 32.8i·19-s + (3.09 + 22.7i)23-s − 5.00·25-s − 36.6·29-s + 41.7·31-s + 25.8·35-s − 12.5i·37-s − 40.5·41-s + 83.7i·43-s + 69.6·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 1.65i·7-s + 0.601i·11-s − 0.334·13-s + 0.914i·17-s − 1.72i·19-s + (0.134 + 0.990i)23-s − 0.200·25-s − 1.26·29-s + 1.34·31-s + 0.738·35-s − 0.339i·37-s − 0.988·41-s + 1.94i·43-s + 1.48·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6638791207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6638791207\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-3.09 - 22.7i)T \) |
| good | 7 | \( 1 - 11.5iT - 49T^{2} \) |
| 11 | \( 1 - 6.61iT - 121T^{2} \) |
| 13 | \( 1 + 4.35T + 169T^{2} \) |
| 17 | \( 1 - 15.5iT - 289T^{2} \) |
| 19 | \( 1 + 32.8iT - 361T^{2} \) |
| 29 | \( 1 + 36.6T + 841T^{2} \) |
| 31 | \( 1 - 41.7T + 961T^{2} \) |
| 37 | \( 1 + 12.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 40.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 83.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 69.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 23.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 45.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 99.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 77.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 52.2T + 5.04e3T^{2} \) |
| 73 | \( 1 - 91.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 11.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 29.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 31.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 110. iT - 9.40e3T^{2} \) |
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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
−8.722189646702142745735307484137, −7.997944250358683382931031229527, −7.20957335455838672669554836850, −6.30644628519769229242655553441, −5.62671924271948868814995888927, −4.97550679235041269890565176762, −4.25239072780781174082988290584, −3.00920868174271870272677501222, −2.32954455015007936546230252725, −1.40135058036446900696898898403,
0.14497762236076118501292301301, 1.06782809545523330223691387234, 2.26399324762648449477858832023, 3.40284614314032997873302456938, 3.89066416603430920705581936117, 4.77641561019401588806608235776, 5.69161144163704968585659288134, 6.57243758915837770256662749384, 7.13948759346418441277848905862, 7.78311494244981563130426551321
