| L(s) = 1 | − 1.70·2-s − 3-s + 0.898·4-s + 0.715·5-s + 1.70·6-s + 1.87·8-s + 9-s − 1.21·10-s + 5.02·11-s − 0.898·12-s + 0.797·13-s − 0.715·15-s − 4.98·16-s − 7.91·17-s − 1.70·18-s − 1.11·19-s + 0.643·20-s − 8.55·22-s + 5.76·23-s − 1.87·24-s − 4.48·25-s − 1.35·26-s − 27-s − 6.92·29-s + 1.21·30-s + 0.0330·31-s + 4.74·32-s + ⋯ |
| L(s) = 1 | − 1.20·2-s − 0.577·3-s + 0.449·4-s + 0.320·5-s + 0.695·6-s + 0.662·8-s + 0.333·9-s − 0.385·10-s + 1.51·11-s − 0.259·12-s + 0.221·13-s − 0.184·15-s − 1.24·16-s − 1.92·17-s − 0.401·18-s − 0.254·19-s + 0.143·20-s − 1.82·22-s + 1.20·23-s − 0.382·24-s − 0.897·25-s − 0.266·26-s − 0.192·27-s − 1.28·29-s + 0.222·30-s + 0.00594·31-s + 0.838·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7057766569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7057766569\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 5 | \( 1 - 0.715T + 5T^{2} \) |
| 11 | \( 1 - 5.02T + 11T^{2} \) |
| 13 | \( 1 - 0.797T + 13T^{2} \) |
| 17 | \( 1 + 7.91T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 - 5.76T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 0.0330T + 31T^{2} \) |
| 37 | \( 1 - 9.85T + 37T^{2} \) |
| 41 | \( 1 + 8.44T + 41T^{2} \) |
| 43 | \( 1 + 4.57T + 43T^{2} \) |
| 47 | \( 1 + 5.76T + 47T^{2} \) |
| 53 | \( 1 + 6.14T + 53T^{2} \) |
| 59 | \( 1 - 0.612T + 59T^{2} \) |
| 67 | \( 1 - 7.19T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 5.69T + 79T^{2} \) |
| 83 | \( 1 - 8.11T + 83T^{2} \) |
| 89 | \( 1 + 4.96T + 89T^{2} \) |
| 97 | \( 1 - 8.89T + 97T^{2} \) |
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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
−7.85757789033059598500848121602, −6.91350649482987968644318578062, −6.65702025225042945526101181917, −5.94601830946868370703644324317, −4.87674320302213941102569936467, −4.35059547827391528276701757370, −3.53030939958183288141106617707, −2.10148243792373262486757721540, −1.56539036001975274112207443122, −0.52613868173688425076075773443,
0.52613868173688425076075773443, 1.56539036001975274112207443122, 2.10148243792373262486757721540, 3.53030939958183288141106617707, 4.35059547827391528276701757370, 4.87674320302213941102569936467, 5.94601830946868370703644324317, 6.65702025225042945526101181917, 6.91350649482987968644318578062, 7.85757789033059598500848121602
