| L(s) = 1 | − 1.74·2-s + 2.44·3-s + 1.05·4-s − 5-s − 4.26·6-s + 1.65·8-s + 2.95·9-s + 1.74·10-s + 11-s + 2.57·12-s − 1.88·13-s − 2.44·15-s − 4.99·16-s + 2.57·17-s − 5.17·18-s − 3.06·19-s − 1.05·20-s − 1.74·22-s − 5.83·23-s + 4.03·24-s + 25-s + 3.28·26-s − 0.0978·27-s − 2.56·29-s + 4.26·30-s − 2.84·31-s + 5.42·32-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 1.40·3-s + 0.526·4-s − 0.447·5-s − 1.74·6-s + 0.584·8-s + 0.986·9-s + 0.552·10-s + 0.301·11-s + 0.742·12-s − 0.522·13-s − 0.630·15-s − 1.24·16-s + 0.623·17-s − 1.21·18-s − 0.703·19-s − 0.235·20-s − 0.372·22-s − 1.21·23-s + 0.824·24-s + 0.200·25-s + 0.645·26-s − 0.0188·27-s − 0.475·29-s + 0.778·30-s − 0.511·31-s + 0.958·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.74T + 2T^{2} \) |
| 3 | \( 1 - 2.44T + 3T^{2} \) |
| 13 | \( 1 + 1.88T + 13T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 + 2.84T + 31T^{2} \) |
| 37 | \( 1 + 4.98T + 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 4.87T + 47T^{2} \) |
| 53 | \( 1 + 1.90T + 53T^{2} \) |
| 59 | \( 1 + 8.36T + 59T^{2} \) |
| 61 | \( 1 - 6.89T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 + 0.100T + 73T^{2} \) |
| 79 | \( 1 - 9.19T + 79T^{2} \) |
| 83 | \( 1 + 7.62T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 1.79T + 97T^{2} \) |
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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
−8.342774655846369139801650771166, −8.095882217034903750964945040384, −7.35759098502735854588603078713, −6.64976069186570711174333922602, −5.28430958625419700961811680439, −4.16135587370416064938913224531, −3.52685501044254371140718428260, −2.36478482456147865483504402533, −1.58433162521783909265236250680, 0,
1.58433162521783909265236250680, 2.36478482456147865483504402533, 3.52685501044254371140718428260, 4.16135587370416064938913224531, 5.28430958625419700961811680439, 6.64976069186570711174333922602, 7.35759098502735854588603078713, 8.095882217034903750964945040384, 8.342774655846369139801650771166
