| L(s) = 1 | − 2-s − 4-s + 4·5-s + 3·8-s − 4·10-s − 11-s + 2·13-s − 16-s − 2·17-s + 6·19-s − 4·20-s + 22-s + 4·23-s + 11·25-s − 2·26-s − 6·29-s − 4·31-s − 5·32-s + 2·34-s − 6·37-s − 6·38-s + 12·40-s + 10·41-s + 6·43-s + 44-s − 4·46-s + 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.78·5-s + 1.06·8-s − 1.26·10-s − 0.301·11-s + 0.554·13-s − 1/4·16-s − 0.485·17-s + 1.37·19-s − 0.894·20-s + 0.213·22-s + 0.834·23-s + 11/5·25-s − 0.392·26-s − 1.11·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s − 0.986·37-s − 0.973·38-s + 1.89·40-s + 1.56·41-s + 0.914·43-s + 0.150·44-s − 0.589·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.786277918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.786277918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.604297009326062010919725959715, −7.49691978365161542587362771144, −7.03813503422928946411270227735, −5.87856949643544530766441944075, −5.52362023628489797607880002401, −4.77832359202318803218455106888, −3.70345922204950928392562748941, −2.60167745622677865777131642937, −1.70466037121130143236373374329, −0.881969017130816614551586580638,
0.881969017130816614551586580638, 1.70466037121130143236373374329, 2.60167745622677865777131642937, 3.70345922204950928392562748941, 4.77832359202318803218455106888, 5.52362023628489797607880002401, 5.87856949643544530766441944075, 7.03813503422928946411270227735, 7.49691978365161542587362771144, 8.604297009326062010919725959715
