L(s) = 1 | − 3-s − 3.20·5-s + 9-s − 4.24·11-s − 3.15·13-s + 3.20·15-s − 4.40·17-s − 3.15·19-s − 4.40·23-s + 5.24·25-s − 27-s + 7.20·29-s − 2.04·31-s + 4.24·33-s − 9.65·37-s + 3.15·39-s − 10.4·41-s − 0.750·43-s − 3.20·45-s + 2.40·47-s + 4.40·51-s − 3.29·53-s + 13.6·55-s + 3.15·57-s − 8.24·59-s − 8.09·61-s + 10.0·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.43·5-s + 0.333·9-s − 1.28·11-s − 0.874·13-s + 0.826·15-s − 1.06·17-s − 0.723·19-s − 0.918·23-s + 1.04·25-s − 0.192·27-s + 1.33·29-s − 0.367·31-s + 0.739·33-s − 1.58·37-s + 0.504·39-s − 1.63·41-s − 0.114·43-s − 0.477·45-s + 0.350·47-s + 0.616·51-s − 0.452·53-s + 1.83·55-s + 0.417·57-s − 1.07·59-s − 1.03·61-s + 1.25·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1231007933\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1231007933\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3.20T + 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 + 3.15T + 13T^{2} \) |
| 17 | \( 1 + 4.40T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 23 | \( 1 + 4.40T + 23T^{2} \) |
| 29 | \( 1 - 7.20T + 29T^{2} \) |
| 31 | \( 1 + 2.04T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 0.750T + 43T^{2} \) |
| 47 | \( 1 - 2.40T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 8.24T + 59T^{2} \) |
| 61 | \( 1 + 8.09T + 61T^{2} \) |
| 67 | \( 1 + 5.15T + 67T^{2} \) |
| 71 | \( 1 + 6.40T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 2.40T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−8.197965102239569869701563417624, −7.59099323980563532332917323499, −6.93285234944250783552437804799, −6.23427234536028422489273452084, −5.07923496502479215161310049106, −4.71549370602128257604884547947, −3.89419253645657089691542175921, −2.95240405114502665272081899105, −1.94788014867139412214005367440, −0.18733929018873429335510484084,
0.18733929018873429335510484084, 1.94788014867139412214005367440, 2.95240405114502665272081899105, 3.89419253645657089691542175921, 4.71549370602128257604884547947, 5.07923496502479215161310049106, 6.23427234536028422489273452084, 6.93285234944250783552437804799, 7.59099323980563532332917323499, 8.197965102239569869701563417624