L(s) = 1 | + (0.951 − 0.309i)3-s + (−0.809 + 0.587i)5-s + 1.61i·7-s + (1.30 − 0.951i)13-s + (−0.587 + 0.809i)15-s + (0.587 + 0.190i)19-s + (0.500 + 1.53i)21-s + (−0.587 + 0.809i)23-s + (0.309 − 0.951i)25-s + (−0.587 + 0.809i)27-s + (−0.190 − 0.587i)29-s + (−0.951 − 0.309i)31-s + (−0.951 − 1.30i)35-s + (0.809 − 0.587i)37-s + (0.951 − 1.30i)39-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)3-s + (−0.809 + 0.587i)5-s + 1.61i·7-s + (1.30 − 0.951i)13-s + (−0.587 + 0.809i)15-s + (0.587 + 0.190i)19-s + (0.500 + 1.53i)21-s + (−0.587 + 0.809i)23-s + (0.309 − 0.951i)25-s + (−0.587 + 0.809i)27-s + (−0.190 − 0.587i)29-s + (−0.951 − 0.309i)31-s + (−0.951 − 1.30i)35-s + (0.809 − 0.587i)37-s + (0.951 − 1.30i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.154274086\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.154274086\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.809 - 0.587i)T \) |
good | 3 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 - 1.61iT - T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + 0.618iT - T^{2} \) |
| 47 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−10.67847091841672608262123516534, −9.422695716015366826204980041786, −8.733580267084066276267827338911, −8.032655075818259882957142100843, −7.49092452083939620616839408903, −6.07861471306836718401753519157, −5.47486571393749890772971815939, −3.76021634913885828515263608812, −3.05764868649847414893347691191, −2.07942221783659488373078447965,
1.26407549998031595804509851284, 3.17325006952746617781311779603, 4.01628998277117597955479134618, 4.46912680491726189403346265781, 6.10188911280424415315988493118, 7.24109474639175735994507933923, 7.85655964919462898957345318787, 8.785042001528410790756469293296, 9.259745161138837675949036030046, 10.41542080954988525464641931699