| L(s) = 1 | + 2-s + 1.41·3-s + 4-s − 1.41·5-s + 1.41·6-s + 8-s − 0.999·9-s − 1.41·10-s − 4·11-s + 1.41·12-s − 1.41·13-s − 2.00·15-s + 16-s + 2.82·17-s − 0.999·18-s − 4.24·19-s − 1.41·20-s − 4·22-s − 4·23-s + 1.41·24-s − 2.99·25-s − 1.41·26-s − 5.65·27-s + 29-s − 2.00·30-s − 1.41·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.816·3-s + 0.5·4-s − 0.632·5-s + 0.577·6-s + 0.353·8-s − 0.333·9-s − 0.447·10-s − 1.20·11-s + 0.408·12-s − 0.392·13-s − 0.516·15-s + 0.250·16-s + 0.685·17-s − 0.235·18-s − 0.973·19-s − 0.316·20-s − 0.852·22-s − 0.834·23-s + 0.288·24-s − 0.599·25-s − 0.277·26-s − 1.08·27-s + 0.185·29-s − 0.365·30-s − 0.254·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 4.24T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2.82T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 2.82T + 97T^{2} \) |
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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
−8.122069160893517061726800899471, −7.83917244483912343527265087574, −7.02407582201295343527011930084, −5.91904419498957307485404317704, −5.30195727711170655609247313250, −4.29510920485375971422504218666, −3.57187365443290026404088366909, −2.75159046292135315041879903557, −2.00087126928217538165123857578, 0,
2.00087126928217538165123857578, 2.75159046292135315041879903557, 3.57187365443290026404088366909, 4.29510920485375971422504218666, 5.30195727711170655609247313250, 5.91904419498957307485404317704, 7.02407582201295343527011930084, 7.83917244483912343527265087574, 8.122069160893517061726800899471
