| L(s) = 1 | + 1.46·3-s + 3.86·5-s − 7-s − 0.860·9-s − 11-s + 13-s + 5.64·15-s − 7.78·17-s − 4.18·19-s − 1.46·21-s − 9.11·23-s + 9.90·25-s − 5.64·27-s + 4.79·29-s − 8.32·31-s − 1.46·33-s − 3.86·35-s − 0.323·37-s + 1.46·39-s + 11.0·41-s − 2.58·43-s − 3.32·45-s + 8.36·47-s + 49-s − 11.3·51-s − 4.98·53-s − 3.86·55-s + ⋯ |
| L(s) = 1 | + 0.844·3-s + 1.72·5-s − 0.377·7-s − 0.286·9-s − 0.301·11-s + 0.277·13-s + 1.45·15-s − 1.88·17-s − 0.959·19-s − 0.319·21-s − 1.90·23-s + 1.98·25-s − 1.08·27-s + 0.890·29-s − 1.49·31-s − 0.254·33-s − 0.652·35-s − 0.0531·37-s + 0.234·39-s + 1.72·41-s − 0.393·43-s − 0.495·45-s + 1.22·47-s + 0.142·49-s − 1.59·51-s − 0.685·53-s − 0.520·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.46T + 3T^{2} \) |
| 5 | \( 1 - 3.86T + 5T^{2} \) |
| 17 | \( 1 + 7.78T + 17T^{2} \) |
| 19 | \( 1 + 4.18T + 19T^{2} \) |
| 23 | \( 1 + 9.11T + 23T^{2} \) |
| 29 | \( 1 - 4.79T + 29T^{2} \) |
| 31 | \( 1 + 8.32T + 31T^{2} \) |
| 37 | \( 1 + 0.323T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 2.58T + 43T^{2} \) |
| 47 | \( 1 - 8.36T + 47T^{2} \) |
| 53 | \( 1 + 4.98T + 53T^{2} \) |
| 59 | \( 1 + 4.97T + 59T^{2} \) |
| 61 | \( 1 - 1.13T + 61T^{2} \) |
| 67 | \( 1 - 1.18T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 1.44T + 73T^{2} \) |
| 79 | \( 1 + 2.33T + 79T^{2} \) |
| 83 | \( 1 + 6.98T + 83T^{2} \) |
| 89 | \( 1 + 2.90T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.51105983994662179256497527844, −6.64415420542117711965058257913, −6.02170469181912200130719556420, −5.71328393309283639644901701633, −4.57558867919856544924769673726, −3.88991272326361457793272365122, −2.74055194943842426213330460557, −2.29796990338717869773494490001, −1.71979115164282027571285931392, 0,
1.71979115164282027571285931392, 2.29796990338717869773494490001, 2.74055194943842426213330460557, 3.88991272326361457793272365122, 4.57558867919856544924769673726, 5.71328393309283639644901701633, 6.02170469181912200130719556420, 6.64415420542117711965058257913, 7.51105983994662179256497527844
