L(s) = 1 | + (0.207 − 0.358i)2-s + (−0.5 + 0.866i)3-s + (0.914 + 1.58i)4-s + 2.82·5-s + (0.207 + 0.358i)6-s + (1.41 + 2.44i)7-s + 1.58·8-s + (−0.499 − 0.866i)9-s + (0.585 − 1.01i)10-s + (−1 + 1.73i)11-s − 1.82·12-s + 1.17·14-s + (−1.41 + 2.44i)15-s + (−1.49 + 2.59i)16-s + (−3.82 − 6.63i)17-s − 0.414·18-s + ⋯ |
L(s) = 1 | + (0.146 − 0.253i)2-s + (−0.288 + 0.499i)3-s + (0.457 + 0.791i)4-s + 1.26·5-s + (0.0845 + 0.146i)6-s + (0.534 + 0.925i)7-s + 0.560·8-s + (−0.166 − 0.288i)9-s + (0.185 − 0.320i)10-s + (−0.301 + 0.522i)11-s − 0.527·12-s + 0.313·14-s + (−0.365 + 0.632i)15-s + (−0.374 + 0.649i)16-s + (−0.928 − 1.60i)17-s − 0.0976·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.70119 + 0.953309i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.70119 + 0.953309i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.207 + 0.358i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + (-1.41 - 2.44i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.82 + 6.63i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + (3.82 - 6.63i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.58 + 4.47i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.828 - 1.43i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + (-3.82 - 6.63i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.65 + 11.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.41 + 5.91i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 0.343T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 3.65T + 83T^{2} \) |
| 89 | \( 1 + (-7.41 + 12.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.82 - 3.16i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.10471092894043419211742644436, −10.29127628754537152267862069135, −9.226798472947713878655754622885, −8.680446717925166289506703244327, −7.29809213740979468354591257644, −6.40940494951528531770607497376, −5.25187016563718318674416825070, −4.54936766426418885587065903167, −2.79537479818816027185796574490, −2.12457780496368629260633669199,
1.30196099758597099383030985737, 2.20169886619449693510770836419, 4.21130956263690406268950429095, 5.54158196755666875524205763952, 6.00879206887661471875555171306, 6.90616231386214811538285367320, 7.85367015335727319243938035844, 9.029087794000838497663895177775, 10.23297020961379057939477395500, 10.65433315101747545523567762092