| L(s) = 1 | + 1.67·3-s + 2.65·5-s + 7-s − 0.202·9-s − 0.826·11-s − 5.57·13-s + 4.44·15-s − 1.74·17-s − 5.93·19-s + 1.67·21-s + 3.04·23-s + 2.05·25-s − 5.35·27-s − 6.26·29-s − 7.90·31-s − 1.38·33-s + 2.65·35-s + 8.30·37-s − 9.32·39-s + 1.38·41-s − 2.45·43-s − 0.538·45-s − 1.80·47-s + 49-s − 2.92·51-s − 13.7·53-s − 2.19·55-s + ⋯ |
| L(s) = 1 | + 0.965·3-s + 1.18·5-s + 0.377·7-s − 0.0675·9-s − 0.249·11-s − 1.54·13-s + 1.14·15-s − 0.424·17-s − 1.36·19-s + 0.364·21-s + 0.634·23-s + 0.411·25-s − 1.03·27-s − 1.16·29-s − 1.42·31-s − 0.240·33-s + 0.448·35-s + 1.36·37-s − 1.49·39-s + 0.216·41-s − 0.375·43-s − 0.0802·45-s − 0.262·47-s + 0.142·49-s − 0.409·51-s − 1.89·53-s − 0.296·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 5 | \( 1 - 2.65T + 5T^{2} \) |
| 11 | \( 1 + 0.826T + 11T^{2} \) |
| 13 | \( 1 + 5.57T + 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 23 | \( 1 - 3.04T + 23T^{2} \) |
| 29 | \( 1 + 6.26T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 - 8.30T + 37T^{2} \) |
| 41 | \( 1 - 1.38T + 41T^{2} \) |
| 43 | \( 1 + 2.45T + 43T^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 - 4.38T + 61T^{2} \) |
| 67 | \( 1 - 6.80T + 67T^{2} \) |
| 71 | \( 1 - 1.11T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 7.61T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 0.428T + 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−7.72476147466774328046738884431, −6.95830716834262147464226239901, −6.16694431657093922559598454345, −5.42043292833173479155262713879, −4.80691319722794956219812243494, −3.90091946606623109185177216421, −2.86658536993053263014555078197, −2.22652272144260970710226719504, −1.79330256869957482332087866209, 0,
1.79330256869957482332087866209, 2.22652272144260970710226719504, 2.86658536993053263014555078197, 3.90091946606623109185177216421, 4.80691319722794956219812243494, 5.42043292833173479155262713879, 6.16694431657093922559598454345, 6.95830716834262147464226239901, 7.72476147466774328046738884431
