| L(s) = 1 | − 2-s + 4-s − 5·7-s − 8-s + 5·14-s + 16-s + 9·17-s − 3·23-s + 2·25-s − 5·28-s − 11·31-s − 32-s − 9·34-s + 6·41-s + 3·46-s + 3·47-s + 7·49-s − 2·50-s + 5·56-s + 11·62-s + 64-s + 9·68-s + 27·71-s + 10·73-s + 7·79-s − 6·82-s − 3·92-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.88·7-s − 0.353·8-s + 1.33·14-s + 1/4·16-s + 2.18·17-s − 0.625·23-s + 2/5·25-s − 0.944·28-s − 1.97·31-s − 0.176·32-s − 1.54·34-s + 0.937·41-s + 0.442·46-s + 0.437·47-s + 49-s − 0.282·50-s + 0.668·56-s + 1.39·62-s + 1/8·64-s + 1.09·68-s + 3.20·71-s + 1.17·73-s + 0.787·79-s − 0.662·82-s − 0.312·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7786963429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7786963429\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−9.614415656704028107435040650054, −9.277475848857048067973801633118, −8.831634565037408267540926373341, −7.991705418809130178175906363427, −7.70990030547262321573258641841, −7.20533631624972393093413350207, −6.49939725047779375985738986397, −6.26973005910684302986456828007, −5.53422693827246838456036024559, −5.18520448262289844865199841549, −3.87862951291467162478218139496, −3.52507738292602038758300222993, −2.97804383877001865980260213278, −2.03995234359922457458096336700, −0.73014561157098743540667847689,
0.73014561157098743540667847689, 2.03995234359922457458096336700, 2.97804383877001865980260213278, 3.52507738292602038758300222993, 3.87862951291467162478218139496, 5.18520448262289844865199841549, 5.53422693827246838456036024559, 6.26973005910684302986456828007, 6.49939725047779375985738986397, 7.20533631624972393093413350207, 7.70990030547262321573258641841, 7.991705418809130178175906363427, 8.831634565037408267540926373341, 9.277475848857048067973801633118, 9.614415656704028107435040650054
