| L(s) = 1 | − 3-s − 5-s + 2·11-s + 8·13-s + 15-s + 2·17-s + 4·19-s − 6·23-s − 4·25-s − 2·27-s + 5·31-s − 2·33-s + 2·37-s − 8·39-s + 7·41-s − 43-s + 6·47-s − 2·51-s − 15·53-s − 2·55-s − 4·57-s + 14·59-s + 9·61-s − 8·65-s + 67-s + 6·69-s + 18·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.603·11-s + 2.21·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s − 1.25·23-s − 4/5·25-s − 0.384·27-s + 0.898·31-s − 0.348·33-s + 0.328·37-s − 1.28·39-s + 1.09·41-s − 0.152·43-s + 0.875·47-s − 0.280·51-s − 2.06·53-s − 0.269·55-s − 0.529·57-s + 1.82·59-s + 1.15·61-s − 0.992·65-s + 0.122·67-s + 0.722·69-s + 2.13·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.681571679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.681571679\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 89 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 81 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 157 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 146 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 137 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 202 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 25 T + 297 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 170 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 245 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−8.598539873270534062537199283794, −8.392727398063302631771492985602, −8.157185473535855845248277709978, −7.77834820253630330623211250396, −7.33373535512831631070207449318, −6.88163862885747332863183352606, −6.48498370050946151082146349990, −6.16561685839038769275965072916, −5.73186334231391871219904263621, −5.62438240955602658525752807987, −5.13728492055850350978206559993, −4.32185421330321881091255475444, −4.20567628109388525966213488019, −3.79037105057957972672210184380, −3.36242987153296577280979426608, −3.00985045279199418994643201030, −2.10687829164353309312528916744, −1.75567212017422970424995226382, −0.903972632988435692391668117711, −0.70641798711397202568177346921,
0.70641798711397202568177346921, 0.903972632988435692391668117711, 1.75567212017422970424995226382, 2.10687829164353309312528916744, 3.00985045279199418994643201030, 3.36242987153296577280979426608, 3.79037105057957972672210184380, 4.20567628109388525966213488019, 4.32185421330321881091255475444, 5.13728492055850350978206559993, 5.62438240955602658525752807987, 5.73186334231391871219904263621, 6.16561685839038769275965072916, 6.48498370050946151082146349990, 6.88163862885747332863183352606, 7.33373535512831631070207449318, 7.77834820253630330623211250396, 8.157185473535855845248277709978, 8.392727398063302631771492985602, 8.598539873270534062537199283794
