| L(s) = 1 | + 2·3-s − 5-s − 3·7-s + 9-s − 3·11-s + 4·13-s − 2·15-s + 5·17-s − 6·21-s − 4·25-s − 4·27-s − 2·29-s − 8·31-s − 6·33-s + 3·35-s + 10·37-s + 8·39-s − 6·41-s − 7·43-s − 45-s − 9·47-s + 2·49-s + 10·51-s + 8·53-s + 3·55-s − 14·59-s − 5·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.904·11-s + 1.10·13-s − 0.516·15-s + 1.21·17-s − 1.30·21-s − 4/5·25-s − 0.769·27-s − 0.371·29-s − 1.43·31-s − 1.04·33-s + 0.507·35-s + 1.64·37-s + 1.28·39-s − 0.937·41-s − 1.06·43-s − 0.149·45-s − 1.31·47-s + 2/7·49-s + 1.40·51-s + 1.09·53-s + 0.404·55-s − 1.82·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.156178009738853762235606038891, −7.965522619530435961346481341902, −7.06922074266953424983972299919, −6.07799811105081656243109859982, −5.42919597536377012060510505576, −4.07655332930450579048686260784, −3.35073146410565689773816100343, −2.96150641784364341581613580420, −1.69229233442531981018945201033, 0,
1.69229233442531981018945201033, 2.96150641784364341581613580420, 3.35073146410565689773816100343, 4.07655332930450579048686260784, 5.42919597536377012060510505576, 6.07799811105081656243109859982, 7.06922074266953424983972299919, 7.965522619530435961346481341902, 8.156178009738853762235606038891
