| L(s) = 1 | − 2.50·3-s − 1.52·5-s − 0.415i·7-s + 3.29·9-s + i·11-s − 4.53i·13-s + 3.83·15-s + 0.133·17-s + (4.33 + 0.450i)19-s + 1.04i·21-s + 3.38i·23-s − 2.66·25-s − 0.743·27-s − 3.02i·29-s − 4.01·31-s + ⋯ |
| L(s) = 1 | − 1.44·3-s − 0.683·5-s − 0.156i·7-s + 1.09·9-s + 0.301i·11-s − 1.25i·13-s + 0.990·15-s + 0.0323·17-s + (0.994 + 0.103i)19-s + 0.227i·21-s + 0.705i·23-s − 0.532·25-s − 0.143·27-s − 0.562i·29-s − 0.720·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5228533273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5228533273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - iT \) |
| 19 | \( 1 + (-4.33 - 0.450i)T \) |
| good | 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 + 1.52T + 5T^{2} \) |
| 7 | \( 1 + 0.415iT - 7T^{2} \) |
| 13 | \( 1 + 4.53iT - 13T^{2} \) |
| 17 | \( 1 - 0.133T + 17T^{2} \) |
| 23 | \( 1 - 3.38iT - 23T^{2} \) |
| 29 | \( 1 + 3.02iT - 29T^{2} \) |
| 31 | \( 1 + 4.01T + 31T^{2} \) |
| 37 | \( 1 + 1.85iT - 37T^{2} \) |
| 41 | \( 1 - 5.83iT - 41T^{2} \) |
| 43 | \( 1 - 7.93iT - 43T^{2} \) |
| 47 | \( 1 - 12.1iT - 47T^{2} \) |
| 53 | \( 1 + 5.96iT - 53T^{2} \) |
| 59 | \( 1 + 7.95T + 59T^{2} \) |
| 61 | \( 1 + 3.03T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 5.95T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 0.959T + 79T^{2} \) |
| 83 | \( 1 + 11.0iT - 83T^{2} \) |
| 89 | \( 1 + 9.52iT - 89T^{2} \) |
| 97 | \( 1 - 9.20iT - 97T^{2} \) |
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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
−8.028304192041360656593280983916, −7.74101601708101224973166159399, −6.88976367902724305249438375384, −5.99970754007376668042739892583, −5.46513924510675887550826823055, −4.74595694083164651644803961263, −3.85674445101164554196406524353, −2.92534296062438607797453087474, −1.32274922572243897558592180548, −0.29054927623883439578050938680,
0.832758780539640386135435379543, 2.13940758351483151251975015710, 3.57155480205694202069341232692, 4.25305667271160498951889307532, 5.21866918770534967456717529586, 5.63700273049126919509753391431, 6.73529377382235287325854423433, 6.98908755403991850511235285281, 7.997072775859733027558438461584, 8.863793853243974905432630630723
