| L(s) = 1 | − 4·5-s − 7-s − 3·9-s + 12·17-s + 2·25-s + 4·35-s − 4·37-s − 4·41-s − 8·43-s + 12·45-s + 16·47-s + 49-s + 3·63-s − 8·67-s + 32·79-s + 9·81-s − 16·83-s − 48·85-s + 12·89-s − 4·101-s − 20·109-s − 12·119-s − 6·121-s + 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.377·7-s − 9-s + 2.91·17-s + 2/5·25-s + 0.676·35-s − 0.657·37-s − 0.624·41-s − 1.21·43-s + 1.78·45-s + 2.33·47-s + 1/7·49-s + 0.377·63-s − 0.977·67-s + 3.60·79-s + 81-s − 1.75·83-s − 5.20·85-s + 1.27·89-s − 0.398·101-s − 1.91·109-s − 1.10·119-s − 0.545·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.772026867484659934825308819852, −8.093537689649372229665039418958, −8.078246359041317001571436271675, −7.40397429472305265033368100557, −7.30264187759595822059178656689, −6.35373904523042773985731435784, −5.92791093661634117809390012383, −5.29340219049011126125230323142, −4.98409920426632935260415692638, −3.89550776474805329328185747620, −3.68340574560716411313233602487, −3.26735667825841356907489548421, −2.50561452898491372410349134801, −1.15037892508990574535308118020, 0,
1.15037892508990574535308118020, 2.50561452898491372410349134801, 3.26735667825841356907489548421, 3.68340574560716411313233602487, 3.89550776474805329328185747620, 4.98409920426632935260415692638, 5.29340219049011126125230323142, 5.92791093661634117809390012383, 6.35373904523042773985731435784, 7.30264187759595822059178656689, 7.40397429472305265033368100557, 8.078246359041317001571436271675, 8.093537689649372229665039418958, 8.772026867484659934825308819852
