| L(s) = 1 | − 3-s − 5-s + (−1.77 + 3.06i)7-s + 9-s + (2.11 − 3.66i)11-s + (−1.76 − 3.06i)13-s + 15-s + (0.838 + 1.45i)17-s + (0.842 + 1.45i)19-s + (1.77 − 3.06i)21-s + (0.974 + 1.68i)23-s + 25-s − 27-s + (−0.599 + 1.03i)29-s + (−0.178 + 0.309i)31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + (−0.669 + 1.15i)7-s + 0.333·9-s + (0.638 − 1.10i)11-s + (−0.490 − 0.848i)13-s + 0.258·15-s + (0.203 + 0.352i)17-s + (0.193 + 0.334i)19-s + (0.386 − 0.669i)21-s + (0.203 + 0.352i)23-s + 0.200·25-s − 0.192·27-s + (−0.111 + 0.192i)29-s + (−0.0320 + 0.0555i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8029776263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8029776263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (6.10 + 5.45i)T \) |
| good | 7 | \( 1 + (1.77 - 3.06i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.11 + 3.66i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.76 + 3.06i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.838 - 1.45i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.842 - 1.45i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.974 - 1.68i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.599 - 1.03i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.178 - 0.309i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.07 - 3.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.74 - 9.95i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 + (-2.77 + 4.80i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 5.36T + 53T^{2} \) |
| 59 | \( 1 - 5.02T + 59T^{2} \) |
| 61 | \( 1 + (6.51 + 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (-3.35 + 5.80i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.14 - 1.97i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.42 + 5.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8.45 + 14.6i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2.17T + 89T^{2} \) |
| 97 | \( 1 + (-0.667 - 1.15i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.317613592171736102235522584861, −7.67002595963930923068164293185, −6.57721409651083426235979915586, −6.13300790370783778381402173288, −5.42432884805348363544760434016, −4.71178903115683705794707035086, −3.36652052508338456770484116449, −3.13510701192979480077422086954, −1.66153378646432683631998566124, −0.33649419101932912634840081955,
0.841688101719534656040257661227, 2.03252728847301507130206902541, 3.31861481355446911311632692929, 4.22239641620583757295213383760, 4.56212095268130161638661240582, 5.60416349499122519980795954406, 6.72230727746270551389235576813, 7.02012819109130341148885772796, 7.48168007375322643615214916506, 8.624281227149326773586303471367
