| L(s) = 1 | − 2·3-s − 5-s + 4·7-s − 3·9-s + 5·11-s − 3·13-s + 2·15-s − 17-s + 5·19-s − 8·21-s + 4·23-s − 8·25-s + 14·27-s + 2·29-s − 10·33-s − 4·35-s + 6·37-s + 6·39-s + 3·45-s − 3·47-s + 3·49-s + 2·51-s + 7·53-s − 5·55-s − 10·57-s + 5·59-s + 7·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1.51·7-s − 9-s + 1.50·11-s − 0.832·13-s + 0.516·15-s − 0.242·17-s + 1.14·19-s − 1.74·21-s + 0.834·23-s − 8/5·25-s + 2.69·27-s + 0.371·29-s − 1.74·33-s − 0.676·35-s + 0.986·37-s + 0.960·39-s + 0.447·45-s − 0.437·47-s + 3/7·49-s + 0.280·51-s + 0.961·53-s − 0.674·55-s − 1.32·57-s + 0.650·59-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2715904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2715904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.564294738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564294738\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 103 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 43 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 57 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 63 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 85 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 123 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 195 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 163 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 134 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 190 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.460080663034889387795759506824, −9.236464544250210766526572231223, −8.635378370698279311082325329107, −8.468415019493519488206893795551, −7.82553564936069844310171972376, −7.80370088671885762117664482968, −6.98691449928033323248517391409, −6.88346937481127247619268164761, −6.24140835907272726191988952457, −5.86768279274327713840896806376, −5.45324636286747665959214037875, −5.13638597931836710709326623845, −4.55772885866834680462626523828, −4.49285702077725582410786317072, −3.52028044741925137428367690924, −3.39368516164756857371869630538, −2.40126078545534586067712670738, −2.04739246025924909042182796049, −1.06526923345006795426192644459, −0.63321708743367686820862594339,
0.63321708743367686820862594339, 1.06526923345006795426192644459, 2.04739246025924909042182796049, 2.40126078545534586067712670738, 3.39368516164756857371869630538, 3.52028044741925137428367690924, 4.49285702077725582410786317072, 4.55772885866834680462626523828, 5.13638597931836710709326623845, 5.45324636286747665959214037875, 5.86768279274327713840896806376, 6.24140835907272726191988952457, 6.88346937481127247619268164761, 6.98691449928033323248517391409, 7.80370088671885762117664482968, 7.82553564936069844310171972376, 8.468415019493519488206893795551, 8.635378370698279311082325329107, 9.236464544250210766526572231223, 9.460080663034889387795759506824
