L(s) = 1 | + 4-s + 3·11-s + 16-s + 12·23-s − 10·25-s − 8·31-s − 8·37-s + 3·44-s + 12·47-s − 10·49-s − 24·53-s − 6·59-s + 64-s + 10·67-s + 24·71-s − 12·89-s + 12·92-s + 10·97-s − 10·100-s + 28·103-s − 12·113-s − 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 0.904·11-s + 1/4·16-s + 2.50·23-s − 2·25-s − 1.43·31-s − 1.31·37-s + 0.452·44-s + 1.75·47-s − 1.42·49-s − 3.29·53-s − 0.781·59-s + 1/8·64-s + 1.22·67-s + 2.84·71-s − 1.27·89-s + 1.25·92-s + 1.01·97-s − 100-s + 2.75·103-s − 1.12·113-s − 0.181·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.462983784\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.462983784\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−7.50395997694989721606241597227, −7.11985565189376005337664653986, −6.68619713403218599725876916726, −6.33800266514324690374051716264, −5.99857165214414943144038732437, −5.34383976963704470462884839275, −5.06687966116029708324180220223, −4.65710913890891472546443799518, −3.83980116686558664178242573817, −3.65076060485342096132488694617, −3.17003897476168898296021928868, −2.57357432097060737204956983168, −1.75073320851877849888240832016, −1.60693251106342657124988453680, −0.58238806619108750440295441942,
0.58238806619108750440295441942, 1.60693251106342657124988453680, 1.75073320851877849888240832016, 2.57357432097060737204956983168, 3.17003897476168898296021928868, 3.65076060485342096132488694617, 3.83980116686558664178242573817, 4.65710913890891472546443799518, 5.06687966116029708324180220223, 5.34383976963704470462884839275, 5.99857165214414943144038732437, 6.33800266514324690374051716264, 6.68619713403218599725876916726, 7.11985565189376005337664653986, 7.50395997694989721606241597227