| L(s) = 1 | − 6·5-s + 7·7-s + 12·11-s − 82·13-s + 30·17-s + 68·19-s − 216·23-s − 89·25-s − 246·29-s − 112·31-s − 42·35-s + 110·37-s + 246·41-s − 172·43-s − 192·47-s + 49·49-s − 558·53-s − 72·55-s − 540·59-s + 110·61-s + 492·65-s + 140·67-s + 840·71-s − 550·73-s + 84·77-s − 208·79-s − 516·83-s + ⋯ |
| L(s) = 1 | − 0.536·5-s + 0.377·7-s + 0.328·11-s − 1.74·13-s + 0.428·17-s + 0.821·19-s − 1.95·23-s − 0.711·25-s − 1.57·29-s − 0.648·31-s − 0.202·35-s + 0.488·37-s + 0.937·41-s − 0.609·43-s − 0.595·47-s + 1/7·49-s − 1.44·53-s − 0.176·55-s − 1.19·59-s + 0.230·61-s + 0.938·65-s + 0.255·67-s + 1.40·71-s − 0.881·73-s + 0.124·77-s − 0.296·79-s − 0.682·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 82 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 68 T + p^{3} T^{2} \) |
| 23 | \( 1 + 216 T + p^{3} T^{2} \) |
| 29 | \( 1 + 246 T + p^{3} T^{2} \) |
| 31 | \( 1 + 112 T + p^{3} T^{2} \) |
| 37 | \( 1 - 110 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 192 T + p^{3} T^{2} \) |
| 53 | \( 1 + 558 T + p^{3} T^{2} \) |
| 59 | \( 1 + 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 110 T + p^{3} T^{2} \) |
| 67 | \( 1 - 140 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 + 550 T + p^{3} T^{2} \) |
| 79 | \( 1 + 208 T + p^{3} T^{2} \) |
| 83 | \( 1 + 516 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1398 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1586 T + p^{3} T^{2} \) |
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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
−11.36769045390804808334607189627, −10.01949038908630846224976367690, −9.364713950783082040274656552290, −7.83925777564651194973630289705, −7.48693625443598578444463711977, −5.94302480272967733000104894711, −4.78931704989477250882495591841, −3.61943208330552797867352883915, −2.00270477810762360828902519984, 0,
2.00270477810762360828902519984, 3.61943208330552797867352883915, 4.78931704989477250882495591841, 5.94302480272967733000104894711, 7.48693625443598578444463711977, 7.83925777564651194973630289705, 9.364713950783082040274656552290, 10.01949038908630846224976367690, 11.36769045390804808334607189627
