| L(s) = 1 | + 3-s + 3·7-s − 2·9-s − 6·13-s − 3·19-s + 3·21-s + 7·25-s − 5·27-s − 31-s + 5·37-s − 6·39-s + 8·43-s + 2·49-s − 3·57-s + 22·61-s − 6·63-s − 5·67-s + 73-s + 7·75-s + 17·79-s + 81-s − 18·91-s − 93-s + 12·97-s + 103-s − 11·109-s + 5·111-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s − 2/3·9-s − 1.66·13-s − 0.688·19-s + 0.654·21-s + 7/5·25-s − 0.962·27-s − 0.179·31-s + 0.821·37-s − 0.960·39-s + 1.21·43-s + 2/7·49-s − 0.397·57-s + 2.81·61-s − 0.755·63-s − 0.610·67-s + 0.117·73-s + 0.808·75-s + 1.91·79-s + 1/9·81-s − 1.88·91-s − 0.103·93-s + 1.21·97-s + 0.0985·103-s − 1.05·109-s + 0.474·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.143805213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.143805213\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.582940289216015026783389892226, −8.088320312592413466959756137205, −7.80114048259235960907836766108, −7.28807545738691720764432058880, −6.89160799184130278107558333769, −6.29432391122082751995475983401, −5.66348402959265695672505031459, −5.17618939255424054600613012376, −4.79705239349155304414489832443, −4.31328556819456571271521132179, −3.65907448929791464094367620133, −2.88887549759831486527654870061, −2.39105001865705172948814635319, −1.97286664564539718550122058145, −0.75418019730163447126072702470,
0.75418019730163447126072702470, 1.97286664564539718550122058145, 2.39105001865705172948814635319, 2.88887549759831486527654870061, 3.65907448929791464094367620133, 4.31328556819456571271521132179, 4.79705239349155304414489832443, 5.17618939255424054600613012376, 5.66348402959265695672505031459, 6.29432391122082751995475983401, 6.89160799184130278107558333769, 7.28807545738691720764432058880, 7.80114048259235960907836766108, 8.088320312592413466959756137205, 8.582940289216015026783389892226
