L(s) = 1 | + (1.20 − 2.09i)2-s + 1.41·3-s + (−1.91 − 3.31i)4-s + (1.91 + 3.31i)5-s + (1.70 − 2.95i)6-s − 4.41·8-s − 0.999·9-s + 9.24·10-s + 3.41·11-s + (−2.70 − 4.68i)12-s + (3.5 − 0.866i)13-s + (2.70 + 4.68i)15-s + (−1.49 + 2.59i)16-s + (−0.0857 − 0.148i)17-s + (−1.20 + 2.09i)18-s − 6·19-s + ⋯ |
L(s) = 1 | + (0.853 − 1.47i)2-s + 0.816·3-s + (−0.957 − 1.65i)4-s + (0.856 + 1.48i)5-s + (0.696 − 1.20i)6-s − 1.56·8-s − 0.333·9-s + 2.92·10-s + 1.02·11-s + (−0.781 − 1.35i)12-s + (0.970 − 0.240i)13-s + (0.698 + 1.21i)15-s + (−0.374 + 0.649i)16-s + (−0.0208 − 0.0360i)17-s + (−0.284 + 0.492i)18-s − 1.37·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.40562 - 2.14606i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.40562 - 2.14606i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-3.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1.20 + 2.09i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + (-1.91 - 3.31i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 17 | \( 1 + (0.0857 + 0.148i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + (0.707 - 1.22i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.91 + 8.51i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.70 + 4.68i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.74 - 6.48i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.91 - 5.04i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.292 - 0.507i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.82 - 6.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.878 + 1.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 9.82T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + (-0.171 + 0.297i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.328 + 0.568i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.12 + 8.87i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + (3.65 - 6.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.58 - 4.47i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.59253689887281135610267757154, −9.755254634117080525154078481506, −9.103958014714995333200176008571, −7.86476604163830065994312762069, −6.34395612266648988756267522015, −5.92426066861154398834585733432, −4.21209000816355882694270484298, −3.42068674476452744574649696285, −2.60290540300076696219005675464, −1.78750080670595596117854158120,
1.75981270760170625448789626429, 3.64390134104786240496905395485, 4.43074281108618830760897749975, 5.49770218290396218485595032626, 6.10602012087559871308793263449, 7.07796977187795315354882140371, 8.355975001985363241282719975136, 8.836208070934769372705977604485, 9.127820506599636065850307301398, 10.70792311023238410506370211823