| L(s) = 1 | − 2·3-s + 5-s + 4·7-s + 9-s − 6·11-s + 13-s − 2·15-s − 6·17-s − 6·19-s − 8·21-s + 2·23-s + 25-s + 4·27-s − 2·29-s − 2·31-s + 12·33-s + 4·35-s + 10·37-s − 2·39-s + 10·41-s − 6·43-s + 45-s + 4·47-s + 9·49-s + 12·51-s + 6·53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.277·13-s − 0.516·15-s − 1.45·17-s − 1.37·19-s − 1.74·21-s + 0.417·23-s + 1/5·25-s + 0.769·27-s − 0.371·29-s − 0.359·31-s + 2.08·33-s + 0.676·35-s + 1.64·37-s − 0.320·39-s + 1.56·41-s − 0.914·43-s + 0.149·45-s + 0.583·47-s + 9/7·49-s + 1.68·51-s + 0.824·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.107494180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107494180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.368740254295442895962482086648, −7.72216804566781580578343937025, −6.84942858331802793041417138527, −6.00080353636214420798695495431, −5.46341248826595151584501917475, −4.77330741500536519888183679795, −4.29306411533331812740793256589, −2.58939279201107512626869366564, −2.00298096018484693451015770628, −0.61667275851107818195824102627,
0.61667275851107818195824102627, 2.00298096018484693451015770628, 2.58939279201107512626869366564, 4.29306411533331812740793256589, 4.77330741500536519888183679795, 5.46341248826595151584501917475, 6.00080353636214420798695495431, 6.84942858331802793041417138527, 7.72216804566781580578343937025, 8.368740254295442895962482086648
