L(s) = 1 | + (0.809 + 0.587i)2-s + (0.941 − 2.89i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (2.46 − 1.79i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−5.08 − 3.69i)9-s + 10-s + (3.28 + 0.432i)11-s + 3.04·12-s + (0.285 + 0.207i)13-s + (0.309 − 0.951i)14-s + (−0.941 − 2.89i)15-s + (−0.809 + 0.587i)16-s + (0.655 − 0.476i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (0.543 − 1.67i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (1.00 − 0.731i)6-s + (−0.116 − 0.359i)7-s + (−0.109 + 0.336i)8-s + (−1.69 − 1.23i)9-s + 0.316·10-s + (0.991 + 0.130i)11-s + 0.879·12-s + (0.0792 + 0.0575i)13-s + (0.0825 − 0.254i)14-s + (−0.243 − 0.748i)15-s + (−0.202 + 0.146i)16-s + (0.159 − 0.115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.02975 - 1.64726i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.02975 - 1.64726i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (-3.28 - 0.432i)T \) |
good | 3 | \( 1 + (-0.941 + 2.89i)T + (-2.42 - 1.76i)T^{2} \) |
| 13 | \( 1 + (-0.285 - 0.207i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.655 + 0.476i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.801 + 2.46i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + (2.95 + 9.08i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.61 - 3.35i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.10 - 3.38i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.55 - 7.85i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 9.49T + 43T^{2} \) |
| 47 | \( 1 + (2.65 - 8.18i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.26 - 1.64i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.08 - 12.5i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.42 + 3.94i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 1.40T + 67T^{2} \) |
| 71 | \( 1 + (-10.4 + 7.59i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.59 + 4.89i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.95 - 2.14i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.15 + 1.56i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 9.63T + 89T^{2} \) |
| 97 | \( 1 + (-7.92 - 5.75i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.927981757749735013123274701604, −8.993804080387226977825381212264, −8.136308419624428942579619683321, −7.47146402934863226198226622366, −6.48665563830501679271678286459, −6.21530421462477562720388336715, −4.76891626280436564411493266458, −3.47792550817614661042012192962, −2.32565971314654979214787992994, −1.13531311212004966933531085778,
2.06849148666818694313496952581, 3.39144639849295473779361998480, 3.80324666017818163036027095528, 4.99179061415866599191073416086, 5.69183696726637058655883645391, 6.79406625349030129816434625721, 8.351141598688486300107670568481, 9.060790285012672216079751090367, 9.852973480555509946584093272429, 10.31618973010560778750703215518