L(s) = 1 | + 2-s + 4-s + 8-s − 11-s − 1.41·13-s + 16-s + 4·17-s − 5.41·19-s − 22-s − 7.65·23-s − 5·25-s − 1.41·26-s + 5.65·29-s + 3.07·31-s + 32-s + 4·34-s + 3.65·37-s − 5.41·38-s + 9.65·41-s − 10·43-s − 44-s − 7.65·46-s − 1.41·47-s − 5·50-s − 1.41·52-s + 3.65·53-s + 5.65·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.353·8-s − 0.301·11-s − 0.392·13-s + 0.250·16-s + 0.970·17-s − 1.24·19-s − 0.213·22-s − 1.59·23-s − 25-s − 0.277·26-s + 1.05·29-s + 0.551·31-s + 0.176·32-s + 0.685·34-s + 0.601·37-s − 0.878·38-s + 1.50·41-s − 1.52·43-s − 0.150·44-s − 1.12·46-s − 0.206·47-s − 0.707·50-s − 0.196·52-s + 0.502·53-s + 0.742·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 3.07T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.29542208757960868442894179899, −6.48213971345871216746309460409, −5.92242279539087911123370123140, −5.35782790963189340005352511643, −4.34411041864422905157916193890, −4.08251615875410683537663548595, −2.98583232802086492150019843370, −2.36392775752372471971761567888, −1.42685369789567281596929366119, 0,
1.42685369789567281596929366119, 2.36392775752372471971761567888, 2.98583232802086492150019843370, 4.08251615875410683537663548595, 4.34411041864422905157916193890, 5.35782790963189340005352511643, 5.92242279539087911123370123140, 6.48213971345871216746309460409, 7.29542208757960868442894179899