| L(s) = 1 | − 2-s − 3.16·3-s + 4-s − 1.61·5-s + 3.16·6-s + 0.0905·7-s − 8-s + 6.99·9-s + 1.61·10-s + 11-s − 3.16·12-s + 0.481·13-s − 0.0905·14-s + 5.09·15-s + 16-s + 0.419·17-s − 6.99·18-s − 1.64·19-s − 1.61·20-s − 0.286·21-s − 22-s + 0.827·23-s + 3.16·24-s − 2.40·25-s − 0.481·26-s − 12.6·27-s + 0.0905·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.82·3-s + 0.5·4-s − 0.720·5-s + 1.29·6-s + 0.0342·7-s − 0.353·8-s + 2.33·9-s + 0.509·10-s + 0.301·11-s − 0.912·12-s + 0.133·13-s − 0.0241·14-s + 1.31·15-s + 0.250·16-s + 0.101·17-s − 1.64·18-s − 0.378·19-s − 0.360·20-s − 0.0624·21-s − 0.213·22-s + 0.172·23-s + 0.645·24-s − 0.481·25-s − 0.0944·26-s − 2.43·27-s + 0.0171·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 - 0.0905T + 7T^{2} \) |
| 13 | \( 1 - 0.481T + 13T^{2} \) |
| 17 | \( 1 - 0.419T + 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 - 0.827T + 23T^{2} \) |
| 29 | \( 1 + 0.561T + 29T^{2} \) |
| 31 | \( 1 + 8.42T + 31T^{2} \) |
| 37 | \( 1 - 1.21T + 37T^{2} \) |
| 41 | \( 1 + 7.62T + 41T^{2} \) |
| 43 | \( 1 - 7.99T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 1.16T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 + 2.12T + 61T^{2} \) |
| 67 | \( 1 - 8.17T + 67T^{2} \) |
| 71 | \( 1 - 3.26T + 71T^{2} \) |
| 73 | \( 1 + 0.788T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 7.46T + 83T^{2} \) |
| 89 | \( 1 + 8.38T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−7.80924416397354006162689709065, −7.22239383844730819397967921153, −6.61262974740256547991933927899, −5.85413343750551943180644263665, −5.25733838607197635823330497328, −4.29236830438444730645406278991, −3.61715714674026910314372821301, −2.02145526524805759381442342324, −0.939758677473102902484201776499, 0,
0.939758677473102902484201776499, 2.02145526524805759381442342324, 3.61715714674026910314372821301, 4.29236830438444730645406278991, 5.25733838607197635823330497328, 5.85413343750551943180644263665, 6.61262974740256547991933927899, 7.22239383844730819397967921153, 7.80924416397354006162689709065
