| L(s) = 1 | − 1.41i·2-s + 1.73i·3-s − 2.00·4-s + (3.16 + 3.86i)5-s + 2.44·6-s − 4.38·7-s + 2.82i·8-s − 2.99·9-s + (5.47 − 4.47i)10-s − 2.89i·11-s − 3.46i·12-s − 1.58i·13-s + 6.20i·14-s + (−6.70 + 5.48i)15-s + 4.00·16-s + 3.92·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (0.633 + 0.773i)5-s + 0.408·6-s − 0.627·7-s + 0.353i·8-s − 0.333·9-s + (0.547 − 0.447i)10-s − 0.263i·11-s − 0.288i·12-s − 0.121i·13-s + 0.443i·14-s + (−0.446 + 0.365i)15-s + 0.250·16-s + 0.230·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9750699098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9750699098\) |
| \(L(2)\) |
|
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 | \( 1 + (-3.16 - 3.86i)T \) |
| 23 | \( 1 + (-22.3 - 5.42i)T \) |
| good | 7 | \( 1 + 4.38T + 49T^{2} \) |
| 11 | \( 1 + 2.89iT - 121T^{2} \) |
| 13 | \( 1 + 1.58iT - 169T^{2} \) |
| 17 | \( 1 - 3.92T + 289T^{2} \) |
| 19 | \( 1 - 23.3iT - 361T^{2} \) |
| 29 | \( 1 + 8.38T + 841T^{2} \) |
| 31 | \( 1 + 32.5T + 961T^{2} \) |
| 37 | \( 1 + 49.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 67.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 16.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 84.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 1.17T + 2.80e3T^{2} \) |
| 59 | \( 1 + 103.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 12.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 125.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 27.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 25.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 25.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 110.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 14.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 63.9T + 9.40e3T^{2} \) |
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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
−10.57961224644620137502804851388, −9.759941409778958397070784430182, −9.254983440550296250604060867969, −8.128328819920553757090439444135, −6.92479402309821593151371397899, −5.92747547768346033668090289792, −5.09280503075387204045862858821, −3.60238335515389503854271862542, −3.10759996308265682029764571176, −1.70371086200310711350386899492,
0.33634876132175831718413851331, 1.82937534726069777745071846610, 3.33823048550450562939342290660, 4.84052308538587433391612883801, 5.48463766053485657355069372230, 6.62926922082927819523442415660, 7.08502639898387895991733091886, 8.328604881666283519205474655689, 9.013655694020377031151714124400, 9.678920266073313335171029782488
