L(s) = 1 | − 2.41·2-s − 2·3-s + 3.82·4-s + 5-s + 4.82·6-s + 0.828·7-s − 4.41·8-s + 9-s − 2.41·10-s − 4.82·11-s − 7.65·12-s − 2·13-s − 1.99·14-s − 2·15-s + 2.99·16-s − 2.82·17-s − 2.41·18-s + 0.828·19-s + 3.82·20-s − 1.65·21-s + 11.6·22-s − 8.82·23-s + 8.82·24-s + 25-s + 4.82·26-s + 4·27-s + 3.17·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s − 1.15·3-s + 1.91·4-s + 0.447·5-s + 1.97·6-s + 0.313·7-s − 1.56·8-s + 0.333·9-s − 0.763·10-s − 1.45·11-s − 2.21·12-s − 0.554·13-s − 0.534·14-s − 0.516·15-s + 0.749·16-s − 0.685·17-s − 0.569·18-s + 0.190·19-s + 0.856·20-s − 0.361·21-s + 2.48·22-s − 1.84·23-s + 1.80·24-s + 0.200·25-s + 0.946·26-s + 0.769·27-s + 0.599·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 0.343T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 - 4.48T + 97T^{2} \) |
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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
−12.09330472206335963966379431081, −11.17398498211377678449527749359, −10.45858470814087163645578370747, −9.707875196311652840487049333439, −8.399942220823804521006680538118, −7.46035497824188566088516866615, −6.23865594298435037206375773972, −5.11685863390931392049616797616, −2.17766922491229158005785092950, 0,
2.17766922491229158005785092950, 5.11685863390931392049616797616, 6.23865594298435037206375773972, 7.46035497824188566088516866615, 8.399942220823804521006680538118, 9.707875196311652840487049333439, 10.45858470814087163645578370747, 11.17398498211377678449527749359, 12.09330472206335963966379431081