L(s) = 1 | − 1.10·3-s + 2.64·5-s − 3.45·7-s − 1.77·9-s − 5.59·11-s + 5.51·13-s − 2.93·15-s − 0.253·17-s + 1.01·19-s + 3.82·21-s − 0.152·23-s + 2.01·25-s + 5.28·27-s − 5.43·29-s + 9.22·31-s + 6.19·33-s − 9.14·35-s + 5.41·37-s − 6.09·39-s − 2.22·41-s + 5.39·43-s − 4.70·45-s + 8.69·47-s + 4.92·49-s + 0.280·51-s − 4.80·53-s − 14.8·55-s + ⋯ |
L(s) = 1 | − 0.638·3-s + 1.18·5-s − 1.30·7-s − 0.591·9-s − 1.68·11-s + 1.52·13-s − 0.756·15-s − 0.0614·17-s + 0.232·19-s + 0.833·21-s − 0.0317·23-s + 0.403·25-s + 1.01·27-s − 1.00·29-s + 1.65·31-s + 1.07·33-s − 1.54·35-s + 0.890·37-s − 0.976·39-s − 0.348·41-s + 0.823·43-s − 0.701·45-s + 1.26·47-s + 0.703·49-s + 0.0392·51-s − 0.659·53-s − 1.99·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + 1.10T + 3T^{2} \) |
| 5 | \( 1 - 2.64T + 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 11 | \( 1 + 5.59T + 11T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 17 | \( 1 + 0.253T + 17T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 + 0.152T + 23T^{2} \) |
| 29 | \( 1 + 5.43T + 29T^{2} \) |
| 31 | \( 1 - 9.22T + 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 - 5.39T + 43T^{2} \) |
| 47 | \( 1 - 8.69T + 47T^{2} \) |
| 53 | \( 1 + 4.80T + 53T^{2} \) |
| 59 | \( 1 - 6.73T + 59T^{2} \) |
| 61 | \( 1 + 9.30T + 61T^{2} \) |
| 67 | \( 1 + 2.64T + 67T^{2} \) |
| 71 | \( 1 + 4.85T + 71T^{2} \) |
| 73 | \( 1 - 4.54T + 73T^{2} \) |
| 79 | \( 1 + 9.05T + 79T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−7.42913832473499730614456857306, −6.32939677630291566016742817615, −6.15179329509245037292732915201, −5.62184634358086722651910357878, −4.95887532860804874308757599240, −3.81920849003154148702773100318, −2.88904031674385753988007389409, −2.45010584902709944934268009768, −1.10526853534821797561606397733, 0,
1.10526853534821797561606397733, 2.45010584902709944934268009768, 2.88904031674385753988007389409, 3.81920849003154148702773100318, 4.95887532860804874308757599240, 5.62184634358086722651910357878, 6.15179329509245037292732915201, 6.32939677630291566016742817615, 7.42913832473499730614456857306