| L(s) = 1 | + 1.70·2-s − 3-s + 0.916·4-s − 1.70·6-s + 0.474·7-s − 1.85·8-s + 9-s − 0.959·11-s − 0.916·12-s − 2.58·13-s + 0.810·14-s − 4.99·16-s + 0.292·17-s + 1.70·18-s + 6.22·19-s − 0.474·21-s − 1.63·22-s + 3.51·23-s + 1.85·24-s − 4.41·26-s − 27-s + 0.434·28-s + 0.124·29-s + 3.47·31-s − 4.82·32-s + 0.959·33-s + 0.499·34-s + ⋯ |
| L(s) = 1 | + 1.20·2-s − 0.577·3-s + 0.458·4-s − 0.697·6-s + 0.179·7-s − 0.654·8-s + 0.333·9-s − 0.289·11-s − 0.264·12-s − 0.716·13-s + 0.216·14-s − 1.24·16-s + 0.0708·17-s + 0.402·18-s + 1.42·19-s − 0.103·21-s − 0.349·22-s + 0.733·23-s + 0.377·24-s − 0.865·26-s − 0.192·27-s + 0.0821·28-s + 0.0230·29-s + 0.624·31-s − 0.852·32-s + 0.167·33-s + 0.0856·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 1.70T + 2T^{2} \) |
| 7 | \( 1 - 0.474T + 7T^{2} \) |
| 11 | \( 1 + 0.959T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 - 0.292T + 17T^{2} \) |
| 19 | \( 1 - 6.22T + 19T^{2} \) |
| 23 | \( 1 - 3.51T + 23T^{2} \) |
| 29 | \( 1 - 0.124T + 29T^{2} \) |
| 31 | \( 1 - 3.47T + 31T^{2} \) |
| 37 | \( 1 + 8.15T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 8.07T + 43T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 4.34T + 59T^{2} \) |
| 61 | \( 1 + 3.86T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 + 2.48T + 71T^{2} \) |
| 73 | \( 1 + 2.09T + 73T^{2} \) |
| 79 | \( 1 - 3.49T + 79T^{2} \) |
| 83 | \( 1 + 6.19T + 83T^{2} \) |
| 89 | \( 1 - 5.84T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−8.068947925042849134357430507327, −7.15863764199974169771286582546, −6.57602314717725538682408008496, −5.67079070801662235725143997916, −4.97272273375839055338044053157, −4.74300958732699116724271333372, −3.47855947485527539727140511094, −2.94072551019035886128782280471, −1.58367607206358413487034261015, 0,
1.58367607206358413487034261015, 2.94072551019035886128782280471, 3.47855947485527539727140511094, 4.74300958732699116724271333372, 4.97272273375839055338044053157, 5.67079070801662235725143997916, 6.57602314717725538682408008496, 7.15863764199974169771286582546, 8.068947925042849134357430507327
