| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s + 3.76i·7-s − i·8-s − 9-s + 3.18·11-s − i·12-s − 6.34i·13-s − 3.76·14-s + 16-s + 2.58i·17-s − i·18-s + 6.34·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.42i·7-s − 0.353i·8-s − 0.333·9-s + 0.959·11-s − 0.288i·12-s − 1.75i·13-s − 1.00·14-s + 0.250·16-s + 0.625i·17-s − 0.235i·18-s + 1.45·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.975047019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975047019\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 3.76iT - 7T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 + 6.34iT - 13T^{2} \) |
| 17 | \( 1 - 2.58iT - 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2.58iT - 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 3.34iT - 47T^{2} \) |
| 53 | \( 1 - 7.16iT - 53T^{2} \) |
| 59 | \( 1 - 4.36T + 59T^{2} \) |
| 61 | \( 1 + 1.52T + 61T^{2} \) |
| 67 | \( 1 + 10.6iT - 67T^{2} \) |
| 71 | \( 1 - 1.81T + 71T^{2} \) |
| 73 | \( 1 - 8.52iT - 73T^{2} \) |
| 79 | \( 1 - 3.18T + 79T^{2} \) |
| 83 | \( 1 + 7.28iT - 83T^{2} \) |
| 89 | \( 1 + 4.58T + 89T^{2} \) |
| 97 | \( 1 + 11.5iT - 97T^{2} \) |
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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
−8.772195834144603294687414131694, −8.144773016521603917502726958420, −7.51012373067509773262676940868, −6.29861415333733870293019587789, −5.83199851976271837554569971680, −5.25359524869819266935286217127, −4.37821124665212795548353055194, −3.32853801070232468896129159873, −2.64509152558798713703371654432, −1.00607036885451214413473720702,
0.815802400370177258680396480171, 1.46684015314601679767745044240, 2.61545389373413665461971020387, 3.74499142717786684657528683130, 4.22144729487272665953416126738, 5.12243972186678582271287021253, 6.31273558169805536753657948973, 7.00512689279514023145080397842, 7.43587507406904908675581344811, 8.406478050177436107450686882872
