| L(s) = 1 | − 2-s − 4-s − 5-s − 2·7-s + 3·8-s + 10-s − 4·11-s + 2·14-s − 16-s + 4·17-s − 6·19-s + 20-s + 4·22-s + 25-s + 2·28-s + 4·29-s + 10·31-s − 5·32-s − 4·34-s + 2·35-s + 2·37-s + 6·38-s − 3·40-s − 6·41-s − 8·43-s + 4·44-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.755·7-s + 1.06·8-s + 0.316·10-s − 1.20·11-s + 0.534·14-s − 1/4·16-s + 0.970·17-s − 1.37·19-s + 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.377·28-s + 0.742·29-s + 1.79·31-s − 0.883·32-s − 0.685·34-s + 0.338·35-s + 0.328·37-s + 0.973·38-s − 0.474·40-s − 0.937·41-s − 1.21·43-s + 0.603·44-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.933566825779465805950857374626, −6.85605070878469029040035317987, −6.38023113852655430033618587580, −5.24704800365555721365905691275, −4.79492102852251063201483324306, −3.88013004305075469922200636586, −3.13338533070239659337291371421, −2.20043355930339332317644484769, −0.892221361851017966542579337510, 0,
0.892221361851017966542579337510, 2.20043355930339332317644484769, 3.13338533070239659337291371421, 3.88013004305075469922200636586, 4.79492102852251063201483324306, 5.24704800365555721365905691275, 6.38023113852655430033618587580, 6.85605070878469029040035317987, 7.933566825779465805950857374626
