| L(s) = 1 | − 2.78·2-s − 1.73·3-s + 5.75·4-s − 2.18·5-s + 4.84·6-s + 2.36·7-s − 10.4·8-s + 0.0237·9-s + 6.09·10-s − 0.884·11-s − 10.0·12-s + 13-s − 6.58·14-s + 3.80·15-s + 17.6·16-s + 2.18·17-s − 0.0662·18-s + 5.05·19-s − 12.6·20-s − 4.11·21-s + 2.46·22-s − 2.02·23-s + 18.1·24-s − 0.208·25-s − 2.78·26-s + 5.17·27-s + 13.6·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 1.00·3-s + 2.87·4-s − 0.978·5-s + 1.97·6-s + 0.894·7-s − 3.69·8-s + 0.00792·9-s + 1.92·10-s − 0.266·11-s − 2.88·12-s + 0.277·13-s − 1.76·14-s + 0.982·15-s + 4.40·16-s + 0.529·17-s − 0.0156·18-s + 1.16·19-s − 2.81·20-s − 0.897·21-s + 0.525·22-s − 0.422·23-s + 3.71·24-s − 0.0417·25-s − 0.546·26-s + 0.995·27-s + 2.57·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.78T + 2T^{2} \) |
| 3 | \( 1 + 1.73T + 3T^{2} \) |
| 5 | \( 1 + 2.18T + 5T^{2} \) |
| 7 | \( 1 - 2.36T + 7T^{2} \) |
| 11 | \( 1 + 0.884T + 11T^{2} \) |
| 17 | \( 1 - 2.18T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 + 2.02T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 + 0.680T + 31T^{2} \) |
| 37 | \( 1 + 7.27T + 37T^{2} \) |
| 41 | \( 1 - 6.76T + 41T^{2} \) |
| 43 | \( 1 + 6.39T + 43T^{2} \) |
| 47 | \( 1 - 1.37T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 + 2.37T + 59T^{2} \) |
| 61 | \( 1 - 2.69T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 7.49T + 73T^{2} \) |
| 79 | \( 1 - 2.06T + 79T^{2} \) |
| 83 | \( 1 + 7.30T + 83T^{2} \) |
| 89 | \( 1 - 5.77T + 89T^{2} \) |
| 97 | \( 1 + 1.09T + 97T^{2} \) |
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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
−7.72569663235503788011216483100, −7.07831664897194577427071100160, −6.36113480511247699257728636311, −5.60713633854568474437437044465, −4.96782661434945662334766393672, −3.65550035957958062365965365732, −2.84055568881584691440903818337, −1.68827433447756381794496306507, −0.892576398002495890161364191818, 0,
0.892576398002495890161364191818, 1.68827433447756381794496306507, 2.84055568881584691440903818337, 3.65550035957958062365965365732, 4.96782661434945662334766393672, 5.60713633854568474437437044465, 6.36113480511247699257728636311, 7.07831664897194577427071100160, 7.72569663235503788011216483100
