| L(s) = 1 | − 3·9-s − 10·11-s + 4·19-s − 14·29-s + 8·31-s − 24·41-s − 49-s + 8·59-s + 8·61-s + 6·79-s + 30·99-s − 36·101-s − 10·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 12·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 9-s − 3.01·11-s + 0.917·19-s − 2.59·29-s + 1.43·31-s − 3.74·41-s − 1/7·49-s + 1.04·59-s + 1.02·61-s + 0.675·79-s + 3.01·99-s − 3.58·101-s − 0.957·109-s + 4.81·121-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 + 0.0773·167-s + 1/13·169-s − 0.917·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2236514327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2236514327\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−9.912945338922367994741340918009, −9.454077248532162809256220197409, −8.835040812046141637034691858472, −8.458500745950601494457365975183, −8.116839195717068430291532311998, −7.83595882214809795956506854805, −7.52092873245584717882288237667, −6.85775665142860311628507074144, −6.65987578942819927692899991328, −5.81848835328526736895234409555, −5.46012397543347808758332835704, −5.15342213409149017338338020428, −5.14392213520221523780011852313, −4.26048787228535012777427976783, −3.48323871218730750841209915753, −3.25862285975149404729416265583, −2.54634699089576332556645417472, −2.36356503599207999051583964960, −1.47810162955999494268069435192, −0.18358137541973819908023094126,
0.18358137541973819908023094126, 1.47810162955999494268069435192, 2.36356503599207999051583964960, 2.54634699089576332556645417472, 3.25862285975149404729416265583, 3.48323871218730750841209915753, 4.26048787228535012777427976783, 5.14392213520221523780011852313, 5.15342213409149017338338020428, 5.46012397543347808758332835704, 5.81848835328526736895234409555, 6.65987578942819927692899991328, 6.85775665142860311628507074144, 7.52092873245584717882288237667, 7.83595882214809795956506854805, 8.116839195717068430291532311998, 8.458500745950601494457365975183, 8.835040812046141637034691858472, 9.454077248532162809256220197409, 9.912945338922367994741340918009
