| L(s) = 1 | + 2·3-s + 5-s + 3·9-s + 4·13-s + 2·15-s + 6·17-s + 4·19-s − 6·23-s + 10·27-s + 12·29-s + 4·31-s − 2·37-s + 8·39-s + 12·41-s − 20·43-s + 3·45-s + 6·47-s + 12·51-s + 6·53-s + 8·57-s − 12·59-s − 2·61-s + 4·65-s − 2·67-s − 12·69-s − 24·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 9-s + 1.10·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s − 1.25·23-s + 1.92·27-s + 2.22·29-s + 0.718·31-s − 0.328·37-s + 1.28·39-s + 1.87·41-s − 3.04·43-s + 0.447·45-s + 0.875·47-s + 1.68·51-s + 0.824·53-s + 1.05·57-s − 1.56·59-s − 0.256·61-s + 0.496·65-s − 0.244·67-s − 1.44·69-s − 2.84·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.558339967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.558339967\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.08148831011459137485763202455, −9.924094709902073761501248050588, −9.308670066528639677666537036848, −8.889299820742350337348374369779, −8.609180564786340551572534074102, −8.053025319550978287116404368811, −7.913492478369081500591229610837, −7.45324651381589791605491044968, −6.65459754637677279776876050756, −6.59858366509367480952496458667, −5.79616924998664611328516655974, −5.67983218297802034433494315335, −4.72867288078326871471190934112, −4.55247514025862520893648744209, −3.82310851333526498030825527579, −3.20179567324733865279262650438, −3.01896128708162683851763325716, −2.36914283912356409762599381318, −1.41411687235421623832430128761, −1.11404015255516090492210452155,
1.11404015255516090492210452155, 1.41411687235421623832430128761, 2.36914283912356409762599381318, 3.01896128708162683851763325716, 3.20179567324733865279262650438, 3.82310851333526498030825527579, 4.55247514025862520893648744209, 4.72867288078326871471190934112, 5.67983218297802034433494315335, 5.79616924998664611328516655974, 6.59858366509367480952496458667, 6.65459754637677279776876050756, 7.45324651381589791605491044968, 7.913492478369081500591229610837, 8.053025319550978287116404368811, 8.609180564786340551572534074102, 8.889299820742350337348374369779, 9.308670066528639677666537036848, 9.924094709902073761501248050588, 10.08148831011459137485763202455
