| L(s) = 1 | − 2·9-s + 2·11-s + 2·29-s − 49-s + 2·71-s − 2·79-s + 3·81-s − 4·99-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2·9-s + 2·11-s + 2·29-s − 49-s + 2·71-s − 2·79-s + 3·81-s − 4·99-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.263456477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.263456477\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.135171382778324257841071363221, −8.693740333959821457089854034731, −8.435144917375820674621288927322, −8.278528101102618437396936855942, −7.77869830947886379175377196948, −7.20956730405814256119774091571, −6.64852555617547412689868432602, −6.63805530292510685727169734775, −6.08377653563261130061383429280, −5.87398772597240078277542629910, −5.35715730028588946782641814349, −4.88573245467035886049476879301, −4.47416567204505404622181264149, −4.04458311518782850130674061207, −3.33801673320425637122490388489, −3.29801930707782801499375189148, −2.64203220024062363062844504018, −2.15721943027365585074170429905, −1.43015267771404055818844501684, −0.76433077461888908712731207369,
0.76433077461888908712731207369, 1.43015267771404055818844501684, 2.15721943027365585074170429905, 2.64203220024062363062844504018, 3.29801930707782801499375189148, 3.33801673320425637122490388489, 4.04458311518782850130674061207, 4.47416567204505404622181264149, 4.88573245467035886049476879301, 5.35715730028588946782641814349, 5.87398772597240078277542629910, 6.08377653563261130061383429280, 6.63805530292510685727169734775, 6.64852555617547412689868432602, 7.20956730405814256119774091571, 7.77869830947886379175377196948, 8.278528101102618437396936855942, 8.435144917375820674621288927322, 8.693740333959821457089854034731, 9.135171382778324257841071363221
