| L(s) = 1 | + 2-s + 2.32·3-s + 4-s − 0.663·5-s + 2.32·6-s − 2.82·7-s + 8-s + 2.41·9-s − 0.663·10-s + 2.32·12-s − 5.19·13-s − 2.82·14-s − 1.54·15-s + 16-s + 4.97·17-s + 2.41·18-s − 2.10·19-s − 0.663·20-s − 6.56·21-s − 1.74·23-s + 2.32·24-s − 4.55·25-s − 5.19·26-s − 1.36·27-s − 2.82·28-s + 5.54·29-s − 1.54·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·3-s + 0.5·4-s − 0.296·5-s + 0.949·6-s − 1.06·7-s + 0.353·8-s + 0.804·9-s − 0.209·10-s + 0.671·12-s − 1.44·13-s − 0.753·14-s − 0.398·15-s + 0.250·16-s + 1.20·17-s + 0.568·18-s − 0.483·19-s − 0.148·20-s − 1.43·21-s − 0.363·23-s + 0.474·24-s − 0.911·25-s − 1.01·26-s − 0.262·27-s − 0.533·28-s + 1.02·29-s − 0.281·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8954 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8954 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 + 0.663T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 13 | \( 1 + 5.19T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 + 2.10T + 19T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 - 8.93T + 31T^{2} \) |
| 41 | \( 1 + 8.22T + 41T^{2} \) |
| 43 | \( 1 + 8.21T + 43T^{2} \) |
| 47 | \( 1 + 6.53T + 47T^{2} \) |
| 53 | \( 1 - 1.75T + 53T^{2} \) |
| 59 | \( 1 + 2.11T + 59T^{2} \) |
| 61 | \( 1 + 6.19T + 61T^{2} \) |
| 67 | \( 1 - 8.01T + 67T^{2} \) |
| 71 | \( 1 + 4.20T + 71T^{2} \) |
| 73 | \( 1 + 5.47T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 8.66T + 83T^{2} \) |
| 89 | \( 1 - 8.85T + 89T^{2} \) |
| 97 | \( 1 - 7.91T + 97T^{2} \) |
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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
−7.42677711008798892871257999807, −6.75332167662225643323127456052, −6.12322901461297660323783248682, −5.17506761869212824652120347916, −4.45029425634356412906801851384, −3.64855496024539897181946032393, −3.04552789979844099140503072324, −2.62495355600381857024309724936, −1.63328682165556191283400976584, 0,
1.63328682165556191283400976584, 2.62495355600381857024309724936, 3.04552789979844099140503072324, 3.64855496024539897181946032393, 4.45029425634356412906801851384, 5.17506761869212824652120347916, 6.12322901461297660323783248682, 6.75332167662225643323127456052, 7.42677711008798892871257999807
