L(s) = 1 | + 1.41i·2-s + 1.73i·3-s − 2.00·4-s + (1.73 + 4.68i)5-s − 2.44·6-s + 7.48·7-s − 2.82i·8-s − 2.99·9-s + (−6.63 + 2.45i)10-s − 2.31i·11-s − 3.46i·12-s + 6.23i·13-s + 10.5i·14-s + (−8.12 + 3.00i)15-s + 4.00·16-s − 2.91·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (0.346 + 0.937i)5-s − 0.408·6-s + 1.06·7-s − 0.353i·8-s − 0.333·9-s + (−0.663 + 0.245i)10-s − 0.210i·11-s − 0.288i·12-s + 0.479i·13-s + 0.756i·14-s + (−0.541 + 0.200i)15-s + 0.250·16-s − 0.171·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.762270697\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.762270697\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + (-1.73 - 4.68i)T \) |
| 23 | \( 1 + (9.64 - 20.8i)T \) |
good | 7 | \( 1 - 7.48T + 49T^{2} \) |
| 11 | \( 1 + 2.31iT - 121T^{2} \) |
| 13 | \( 1 - 6.23iT - 169T^{2} \) |
| 17 | \( 1 + 2.91T + 289T^{2} \) |
| 19 | \( 1 - 25.2iT - 361T^{2} \) |
| 29 | \( 1 - 42.7T + 841T^{2} \) |
| 31 | \( 1 - 20.9T + 961T^{2} \) |
| 37 | \( 1 + 62.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 41.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 38.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 31.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 22.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 17.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 77.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 134.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 4.53iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 28.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 19.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 18.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 145.T + 9.40e3T^{2} \) |
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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
−10.41188910644802300315188310428, −9.989952991635146212678837485294, −8.804749860703855956088811516162, −8.091395501828011760430613032192, −7.16923324676514872023333281297, −6.19612770655216872478054387803, −5.37812807136701283753851700899, −4.36099886168601503843900080639, −3.30901514159792490460406709214, −1.76931008685868433033329685063,
0.64528213391309938384261595406, 1.70888008290919629416344817832, 2.76761018193084637623312394697, 4.50158758022452585381857589845, 4.97608021385119121609750859980, 6.14852572217405355576938259444, 7.38158248233145126979616759282, 8.497589528487560030278655462755, 8.705441485302490272993278446716, 9.970144974966702472384239017774