| L(s) = 1 | − 2·3-s + 3·4-s + 9-s − 6·12-s + 5·16-s + 12·23-s − 6·25-s + 4·27-s − 2·31-s + 3·36-s + 12·37-s − 6·47-s − 10·48-s − 8·49-s + 18·53-s + 3·64-s − 10·67-s − 24·69-s − 6·71-s + 12·75-s − 11·81-s + 6·89-s + 36·92-s + 4·93-s + 4·97-s − 18·100-s − 8·103-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3/2·4-s + 1/3·9-s − 1.73·12-s + 5/4·16-s + 2.50·23-s − 6/5·25-s + 0.769·27-s − 0.359·31-s + 1/2·36-s + 1.97·37-s − 0.875·47-s − 1.44·48-s − 8/7·49-s + 2.47·53-s + 3/8·64-s − 1.22·67-s − 2.88·69-s − 0.712·71-s + 1.38·75-s − 1.22·81-s + 0.635·89-s + 3.75·92-s + 0.414·93-s + 0.406·97-s − 9/5·100-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.544738568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.544738568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.391123376615524289634214778394, −8.988163437873413465273261306041, −8.301479934497777032075454779455, −7.70965142943847156216320356415, −7.23712065372103626215323618313, −6.86050060520135022250260291593, −6.38936219009129754588503267840, −5.88246969543650610358824802805, −5.52973647353527519475543324561, −4.86073842749873988463326706974, −4.27394988321841243924919372259, −3.24666884694853708905752948824, −2.79859931977348035552662373493, −1.93572664561895893044784704738, −0.944613204620489295494413365596,
0.944613204620489295494413365596, 1.93572664561895893044784704738, 2.79859931977348035552662373493, 3.24666884694853708905752948824, 4.27394988321841243924919372259, 4.86073842749873988463326706974, 5.52973647353527519475543324561, 5.88246969543650610358824802805, 6.38936219009129754588503267840, 6.86050060520135022250260291593, 7.23712065372103626215323618313, 7.70965142943847156216320356415, 8.301479934497777032075454779455, 8.988163437873413465273261306041, 9.391123376615524289634214778394
