| L(s) = 1 | − 7-s + 2·9-s − 8·11-s + 4·23-s + 2·25-s + 6·29-s + 14·37-s + 16·43-s + 49-s − 6·53-s − 2·63-s + 4·67-s − 12·71-s + 8·77-s − 12·79-s − 5·81-s − 16·99-s − 12·107-s − 2·109-s − 4·113-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 2/3·9-s − 2.41·11-s + 0.834·23-s + 2/5·25-s + 1.11·29-s + 2.30·37-s + 2.43·43-s + 1/7·49-s − 0.824·53-s − 0.251·63-s + 0.488·67-s − 1.42·71-s + 0.911·77-s − 1.35·79-s − 5/9·81-s − 1.60·99-s − 1.16·107-s − 0.191·109-s − 0.376·113-s + 2.72·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.614486868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614486868\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
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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
−7.908249926347401063781938917468, −7.54634985410732308215919462967, −7.16541948898908520877385477121, −6.74522715721039121033611780550, −6.06013771052747207381420914622, −5.81889940156080348962441416230, −5.30751819332477978849119585401, −4.73606730419091083344916044189, −4.48906628033109933047634427959, −3.94260664344926581948847070989, −3.05810967901473174447951274924, −2.66412835822555006679034914923, −2.50599395392036121865396417525, −1.37002694204899079331471140717, −0.57606156523066974863554768130,
0.57606156523066974863554768130, 1.37002694204899079331471140717, 2.50599395392036121865396417525, 2.66412835822555006679034914923, 3.05810967901473174447951274924, 3.94260664344926581948847070989, 4.48906628033109933047634427959, 4.73606730419091083344916044189, 5.30751819332477978849119585401, 5.81889940156080348962441416230, 6.06013771052747207381420914622, 6.74522715721039121033611780550, 7.16541948898908520877385477121, 7.54634985410732308215919462967, 7.908249926347401063781938917468
