L(s) = 1 | + (−1 + 1.73i)3-s + (−0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (−2 + 3.46i)11-s + 2·13-s + 1.99·15-s + (−1 − 1.73i)19-s + (2 + 3.46i)23-s + (−0.499 + 0.866i)25-s − 4.00·27-s + 10·29-s + (−2 + 3.46i)31-s + (−3.99 − 6.92i)33-s + (1 + 1.73i)37-s + (−2 + 3.46i)39-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.999i)3-s + (−0.223 − 0.387i)5-s + (−0.166 − 0.288i)9-s + (−0.603 + 1.04i)11-s + 0.554·13-s + 0.516·15-s + (−0.229 − 0.397i)19-s + (0.417 + 0.722i)23-s + (−0.0999 + 0.173i)25-s − 0.769·27-s + 1.85·29-s + (−0.359 + 0.622i)31-s + (−0.696 − 1.20i)33-s + (0.164 + 0.284i)37-s + (−0.320 + 0.554i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5881875967\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5881875967\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + (4 + 6.92i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.737164637164440922460775479793, −8.894560463961169330410621558776, −8.133836720361265078082033428551, −7.21131493635429122891472952457, −6.36363319755879462400237017109, −5.19163953181779158949396316990, −4.88099877546702950533352616469, −4.04699960680343292029668420016, −2.98945410855689800002583134443, −1.55012403627239419765466950027,
0.23924823840989363834036037935, 1.40942627623808003791232793947, 2.71905306366609883607589813878, 3.62644502470734658448130307549, 4.84032235104455141887920943262, 5.85114260334268851153971226231, 6.43242723721581732518618445923, 7.02395400974049047387764823666, 8.094931948394374216804053641383, 8.394103003709783179871406074336