| L(s) = 1 | − 4-s + 4·5-s − 9-s − 4·11-s + 16-s + 8·19-s − 4·20-s + 11·25-s + 16·29-s + 36-s + 20·41-s + 4·44-s − 4·45-s + 10·49-s − 16·55-s − 4·59-s + 12·61-s − 64-s − 24·71-s − 8·76-s + 16·79-s + 4·80-s + 81-s − 28·89-s + 32·95-s + 4·99-s − 11·100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s − 1/3·9-s − 1.20·11-s + 1/4·16-s + 1.83·19-s − 0.894·20-s + 11/5·25-s + 2.97·29-s + 1/6·36-s + 3.12·41-s + 0.603·44-s − 0.596·45-s + 10/7·49-s − 2.15·55-s − 0.520·59-s + 1.53·61-s − 1/8·64-s − 2.84·71-s − 0.917·76-s + 1.80·79-s + 0.447·80-s + 1/9·81-s − 2.96·89-s + 3.28·95-s + 0.402·99-s − 1.09·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.951083749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.951083749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.971073844551332353811681872502, −9.749725404846743207279226353934, −9.160381611053136683879904121489, −9.060180545722797218770930246852, −8.352491084350752996434046496005, −8.211742067526369579884699621877, −7.39301760455058688424372193608, −7.32869297130414834726423199471, −6.58168453481228483765175314013, −6.12167642077108184918041839403, −5.70138181127599563863761848194, −5.47728819040868132351069304226, −4.96806792837512902741940079197, −4.59307052288847856892768091564, −3.96482606899410842392394143289, −3.03102283596293859115236733098, −2.58009415572545439057528152636, −2.56216748649858988553186066434, −1.31811950314360201096810550197, −0.869201337974042471442074666886,
0.869201337974042471442074666886, 1.31811950314360201096810550197, 2.56216748649858988553186066434, 2.58009415572545439057528152636, 3.03102283596293859115236733098, 3.96482606899410842392394143289, 4.59307052288847856892768091564, 4.96806792837512902741940079197, 5.47728819040868132351069304226, 5.70138181127599563863761848194, 6.12167642077108184918041839403, 6.58168453481228483765175314013, 7.32869297130414834726423199471, 7.39301760455058688424372193608, 8.211742067526369579884699621877, 8.352491084350752996434046496005, 9.060180545722797218770930246852, 9.160381611053136683879904121489, 9.749725404846743207279226353934, 9.971073844551332353811681872502
