L(s) = 1 | + 1.56·3-s − 2.56·5-s + 7-s − 0.561·9-s − 0.438·11-s + 13-s − 4·15-s − 4·17-s − 5.68·19-s + 1.56·21-s − 5·23-s + 1.56·25-s − 5.56·27-s + 1.43·29-s + 2.12·31-s − 0.684·33-s − 2.56·35-s + 9.56·37-s + 1.56·39-s − 4.43·41-s − 8.80·43-s + 1.43·45-s − 3·47-s + 49-s − 6.24·51-s + 3.68·53-s + 1.12·55-s + ⋯ |
L(s) = 1 | + 0.901·3-s − 1.14·5-s + 0.377·7-s − 0.187·9-s − 0.132·11-s + 0.277·13-s − 1.03·15-s − 0.970·17-s − 1.30·19-s + 0.340·21-s − 1.04·23-s + 0.312·25-s − 1.07·27-s + 0.267·29-s + 0.381·31-s − 0.119·33-s − 0.432·35-s + 1.57·37-s + 0.250·39-s − 0.693·41-s − 1.34·43-s + 0.214·45-s − 0.437·47-s + 0.142·49-s − 0.874·51-s + 0.506·53-s + 0.151·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 + 2.56T + 5T^{2} \) |
| 11 | \( 1 + 0.438T + 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 5.68T + 19T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 - 1.43T + 29T^{2} \) |
| 31 | \( 1 - 2.12T + 31T^{2} \) |
| 37 | \( 1 - 9.56T + 37T^{2} \) |
| 41 | \( 1 + 4.43T + 41T^{2} \) |
| 43 | \( 1 + 8.80T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 3.68T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 + 1.31T + 61T^{2} \) |
| 67 | \( 1 + 0.438T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 5.24T + 73T^{2} \) |
| 79 | \( 1 + 1.87T + 79T^{2} \) |
| 83 | \( 1 + 1.43T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.759774422659961718313261013716, −8.329194135989635518309146726033, −7.83242928568589372738563349646, −6.82873814245696511067497105383, −5.89349308472239864370810495558, −4.52345102105399982072897380581, −4.00253901634024293960404507661, −2.96249644123746485315291530525, −1.95949078335090979603931058254, 0,
1.95949078335090979603931058254, 2.96249644123746485315291530525, 4.00253901634024293960404507661, 4.52345102105399982072897380581, 5.89349308472239864370810495558, 6.82873814245696511067497105383, 7.83242928568589372738563349646, 8.329194135989635518309146726033, 8.759774422659961718313261013716