| L(s) = 1 | + 8·11-s − 16·19-s − 4·29-s + 8·31-s + 20·41-s + 10·49-s + 4·61-s + 24·71-s + 16·79-s − 12·89-s − 28·101-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 3.67·19-s − 0.742·29-s + 1.43·31-s + 3.12·41-s + 10/7·49-s + 0.512·61-s + 2.84·71-s + 1.80·79-s − 1.27·89-s − 2.78·101-s + 2.68·109-s + 2.36·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.962812724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.962812724\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.185989117148983439655265848516, −7.86340149941843290348118159285, −7.26759385787398314979198485766, −6.95438309295676997830431919735, −6.55858142983195531957121901233, −6.48584645820442027954098910832, −6.08669307083478761465946807948, −5.82091162209081564882910286702, −5.38280733985880451136705037685, −4.66702181980443557222511623626, −4.33367976419795829063401444198, −4.24806437155746553539322930066, −3.76720496821210600159941164953, −3.70308177914099961039803022630, −2.72952836326600270498315899350, −2.51328224577348842782040932230, −1.92925336763468307235691418587, −1.73985117507613662361081016545, −0.804191529473431747094285990407, −0.64922751759054705806245418579,
0.64922751759054705806245418579, 0.804191529473431747094285990407, 1.73985117507613662361081016545, 1.92925336763468307235691418587, 2.51328224577348842782040932230, 2.72952836326600270498315899350, 3.70308177914099961039803022630, 3.76720496821210600159941164953, 4.24806437155746553539322930066, 4.33367976419795829063401444198, 4.66702181980443557222511623626, 5.38280733985880451136705037685, 5.82091162209081564882910286702, 6.08669307083478761465946807948, 6.48584645820442027954098910832, 6.55858142983195531957121901233, 6.95438309295676997830431919735, 7.26759385787398314979198485766, 7.86340149941843290348118159285, 8.185989117148983439655265848516
