L(s) = 1 | − 1.41·3-s + 5-s + 7-s − 0.999·9-s − 11-s − 3.41·13-s − 1.41·15-s + 2.24·17-s + 5.65·19-s − 1.41·21-s − 2·23-s + 25-s + 5.65·27-s − 3.17·29-s − 4.58·31-s + 1.41·33-s + 35-s − 2.82·37-s + 4.82·39-s + 4.24·41-s − 2·43-s − 0.999·45-s − 8.24·47-s + 49-s − 3.17·51-s − 4.48·53-s − 55-s + ⋯ |
L(s) = 1 | − 0.816·3-s + 0.447·5-s + 0.377·7-s − 0.333·9-s − 0.301·11-s − 0.946·13-s − 0.365·15-s + 0.543·17-s + 1.29·19-s − 0.308·21-s − 0.417·23-s + 0.200·25-s + 1.08·27-s − 0.588·29-s − 0.823·31-s + 0.246·33-s + 0.169·35-s − 0.464·37-s + 0.773·39-s + 0.662·41-s − 0.304·43-s − 0.149·45-s − 1.20·47-s + 0.142·49-s − 0.444·51-s − 0.616·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 - 4.58T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 9.31T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 7.17T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−8.193631624984522095912602930911, −7.56746594156026595422499980785, −6.78221351977243074473838414069, −5.83972799221896670623650145800, −5.32444561719744107306567782388, −4.77609979247452405366836822985, −3.49022316739119704402417061734, −2.54333878996510077890761010787, −1.39241240053295976483392125158, 0,
1.39241240053295976483392125158, 2.54333878996510077890761010787, 3.49022316739119704402417061734, 4.77609979247452405366836822985, 5.32444561719744107306567782388, 5.83972799221896670623650145800, 6.78221351977243074473838414069, 7.56746594156026595422499980785, 8.193631624984522095912602930911