| L(s) = 1 | − 4-s − 4·11-s + 16-s − 10·29-s + 14·31-s − 14·41-s + 4·44-s − 49-s − 10·59-s − 6·61-s − 64-s − 24·71-s − 20·79-s − 20·89-s − 24·101-s + 10·116-s − 10·121-s − 14·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 14·164-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.20·11-s + 1/4·16-s − 1.85·29-s + 2.51·31-s − 2.18·41-s + 0.603·44-s − 1/7·49-s − 1.30·59-s − 0.768·61-s − 1/8·64-s − 2.84·71-s − 2.25·79-s − 2.11·89-s − 2.38·101-s + 0.928·116-s − 0.909·121-s − 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 1.09·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05742922833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05742922833\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.883989964723449463694998060460, −8.361993452248401674332663346890, −8.271283347321822959936361704402, −7.66073612755731953698203906627, −7.59487215809284527637507476696, −6.94207816007830559333098286809, −6.75169138636964246514282919024, −6.07461372182509048143826465055, −5.84186049145859006025153687105, −5.43108908830618350397645807681, −5.03052056665136633938027836221, −4.56536236662708124509968998309, −4.35545661260127612915762174986, −3.76247462262771287891751462099, −3.24293870197874569928238469942, −2.76336268455975342400461868637, −2.55795096240255478026131980757, −1.52567084440777478986302697144, −1.40496130894693070090633102398, −0.07440349151446252311737012710,
0.07440349151446252311737012710, 1.40496130894693070090633102398, 1.52567084440777478986302697144, 2.55795096240255478026131980757, 2.76336268455975342400461868637, 3.24293870197874569928238469942, 3.76247462262771287891751462099, 4.35545661260127612915762174986, 4.56536236662708124509968998309, 5.03052056665136633938027836221, 5.43108908830618350397645807681, 5.84186049145859006025153687105, 6.07461372182509048143826465055, 6.75169138636964246514282919024, 6.94207816007830559333098286809, 7.59487215809284527637507476696, 7.66073612755731953698203906627, 8.271283347321822959936361704402, 8.361993452248401674332663346890, 8.883989964723449463694998060460
