| L(s) = 1 | − 2-s − 1.71·3-s + 4-s + 1.71·6-s + 2.77·7-s − 8-s − 0.0597·9-s + 2.77·11-s − 1.71·12-s − 5.67·13-s − 2.77·14-s + 16-s − 5.15·17-s + 0.0597·18-s + 1.41·19-s − 4.75·21-s − 2.77·22-s + 0.654·23-s + 1.71·24-s + 5.67·26-s + 5.24·27-s + 2.77·28-s + 4.09·29-s − 7.12·31-s − 32-s − 4.75·33-s + 5.15·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.989·3-s + 0.5·4-s + 0.700·6-s + 1.04·7-s − 0.353·8-s − 0.0199·9-s + 0.836·11-s − 0.494·12-s − 1.57·13-s − 0.741·14-s + 0.250·16-s − 1.25·17-s + 0.0140·18-s + 0.324·19-s − 1.03·21-s − 0.591·22-s + 0.136·23-s + 0.350·24-s + 1.11·26-s + 1.00·27-s + 0.524·28-s + 0.760·29-s − 1.28·31-s − 0.176·32-s − 0.828·33-s + 0.884·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.71T + 3T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 - 2.77T + 11T^{2} \) |
| 13 | \( 1 + 5.67T + 13T^{2} \) |
| 17 | \( 1 + 5.15T + 17T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 - 0.654T + 23T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 - 9.10T + 41T^{2} \) |
| 43 | \( 1 + 9.24T + 43T^{2} \) |
| 47 | \( 1 - 2.77T + 47T^{2} \) |
| 53 | \( 1 - 0.526T + 53T^{2} \) |
| 59 | \( 1 + 3.78T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 4.21T + 73T^{2} \) |
| 79 | \( 1 + 9.90T + 79T^{2} \) |
| 83 | \( 1 + 4.67T + 83T^{2} \) |
| 89 | \( 1 + 9.18T + 89T^{2} \) |
| 97 | \( 1 - 0.0901T + 97T^{2} \) |
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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
−9.252806371056128745338293918818, −8.597933624943370217575593103608, −7.56728795946096006313300444150, −6.91675873748421441549073896911, −6.01918702803850978218887019944, −5.06551071316823688534957987641, −4.38342766702250711591811349920, −2.68508240784627668660706639485, −1.50222857929988507270201049037, 0,
1.50222857929988507270201049037, 2.68508240784627668660706639485, 4.38342766702250711591811349920, 5.06551071316823688534957987641, 6.01918702803850978218887019944, 6.91675873748421441549073896911, 7.56728795946096006313300444150, 8.597933624943370217575593103608, 9.252806371056128745338293918818
