| L(s) = 1 | − 3-s − 4-s + 9-s + 12-s − 13-s + 16-s − 25-s − 27-s − 36-s + 39-s + 2·43-s − 48-s + 49-s + 52-s − 2·61-s − 64-s + 75-s − 2·79-s + 81-s + 100-s + 2·103-s + 108-s − 117-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s + 9-s + 12-s − 13-s + 16-s − 25-s − 27-s − 36-s + 39-s + 2·43-s − 48-s + 49-s + 52-s − 2·61-s − 64-s + 75-s − 2·79-s + 81-s + 100-s + 2·103-s + 108-s − 117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2913537228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2913537228\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−16.89615994646558642421555820819, −15.51291484878802936574704349484, −14.14821592234820384952611063785, −12.89937159831719324582676854317, −11.93271353025467403728401266104, −10.42608525426608152464833032909, −9.334261916221688779645422148440, −7.54499009544086055036625670577, −5.72936210305730426417401616626, −4.39582175376675801531882680457,
4.39582175376675801531882680457, 5.72936210305730426417401616626, 7.54499009544086055036625670577, 9.334261916221688779645422148440, 10.42608525426608152464833032909, 11.93271353025467403728401266104, 12.89937159831719324582676854317, 14.14821592234820384952611063785, 15.51291484878802936574704349484, 16.89615994646558642421555820819
