| L(s) = 1 | + 1.66·2-s + 3-s + 0.782·4-s + 4.32·5-s + 1.66·6-s − 2.03·8-s + 9-s + 7.21·10-s + 5.84·11-s + 0.782·12-s + 4.56·13-s + 4.32·15-s − 4.95·16-s + 1.48·17-s + 1.66·18-s − 7.25·19-s + 3.38·20-s + 9.75·22-s + 3.99·23-s − 2.03·24-s + 13.6·25-s + 7.60·26-s + 27-s − 3.99·29-s + 7.21·30-s − 3.33·31-s − 4.19·32-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 0.577·3-s + 0.391·4-s + 1.93·5-s + 0.680·6-s − 0.718·8-s + 0.333·9-s + 2.28·10-s + 1.76·11-s + 0.225·12-s + 1.26·13-s + 1.11·15-s − 1.23·16-s + 0.360·17-s + 0.393·18-s − 1.66·19-s + 0.756·20-s + 2.08·22-s + 0.832·23-s − 0.414·24-s + 2.73·25-s + 1.49·26-s + 0.192·27-s − 0.742·29-s + 1.31·30-s − 0.599·31-s − 0.742·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.184065130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.184065130\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.66T + 2T^{2} \) |
| 5 | \( 1 - 4.32T + 5T^{2} \) |
| 11 | \( 1 - 5.84T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 - 1.48T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 - 3.99T + 23T^{2} \) |
| 29 | \( 1 + 3.99T + 29T^{2} \) |
| 31 | \( 1 + 3.33T + 31T^{2} \) |
| 37 | \( 1 - 0.681T + 37T^{2} \) |
| 43 | \( 1 + 7.66T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 - 0.715T + 59T^{2} \) |
| 61 | \( 1 - 3.33T + 61T^{2} \) |
| 67 | \( 1 + 7.99T + 67T^{2} \) |
| 71 | \( 1 - 7.51T + 71T^{2} \) |
| 73 | \( 1 + 0.453T + 73T^{2} \) |
| 79 | \( 1 + 6.22T + 79T^{2} \) |
| 83 | \( 1 + 7.75T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
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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
−8.328941342506777299237022507695, −6.70860010846850625818754567512, −6.59556360372128840204687013465, −5.91192888279322133732520731719, −5.27200956625226922458654371612, −4.35878621443967075979020064664, −3.70311655176318088951502799959, −2.95487372908940787764055850383, −1.94076791486321292222352094319, −1.35347719470437705792573388618,
1.35347719470437705792573388618, 1.94076791486321292222352094319, 2.95487372908940787764055850383, 3.70311655176318088951502799959, 4.35878621443967075979020064664, 5.27200956625226922458654371612, 5.91192888279322133732520731719, 6.59556360372128840204687013465, 6.70860010846850625818754567512, 8.328941342506777299237022507695
