| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 3·7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 3·14-s + 15-s + 16-s − 3·17-s − 18-s − 2·19-s + 20-s + 3·21-s + 22-s − 24-s − 4·25-s − 26-s + 27-s + 3·28-s + 29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.654·21-s + 0.213·22-s − 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.566·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.38158709052339, −14.69930613151087, −14.14777635339004, −13.83010092751125, −13.08044518632792, −12.68975726681854, −11.96384800006657, −11.24480088048328, −11.01286461071275, −10.41265267459051, −9.811947120820367, −9.204699632859575, −8.815971453979112, −8.226449840153983, −7.727914894482522, −7.332856130217603, −6.490311397084923, −5.959672320265010, −5.286219659932787, −4.516288790641897, −4.022877052393382, −3.068346040470867, −2.343215942573697, −1.837341415138677, −1.204301875248323, 0,
1.204301875248323, 1.837341415138677, 2.343215942573697, 3.068346040470867, 4.022877052393382, 4.516288790641897, 5.286219659932787, 5.959672320265010, 6.490311397084923, 7.332856130217603, 7.727914894482522, 8.226449840153983, 8.815971453979112, 9.204699632859575, 9.811947120820367, 10.41265267459051, 11.01286461071275, 11.24480088048328, 11.96384800006657, 12.68975726681854, 13.08044518632792, 13.83010092751125, 14.14777635339004, 14.69930613151087, 15.38158709052339
