| L(s) = 1 | + 3-s + 6·11-s − 4·13-s − 4·19-s + 6·23-s + 5·25-s − 27-s + 12·29-s + 8·31-s + 6·33-s − 2·37-s − 4·39-s − 24·41-s − 8·43-s + 12·47-s + 6·53-s − 4·57-s − 10·61-s − 8·67-s + 6·69-s + 12·71-s − 10·73-s + 5·75-s + 4·79-s − 81-s + 24·83-s + 12·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.80·11-s − 1.10·13-s − 0.917·19-s + 1.25·23-s + 25-s − 0.192·27-s + 2.22·29-s + 1.43·31-s + 1.04·33-s − 0.328·37-s − 0.640·39-s − 3.74·41-s − 1.21·43-s + 1.75·47-s + 0.824·53-s − 0.529·57-s − 1.28·61-s − 0.977·67-s + 0.722·69-s + 1.42·71-s − 1.17·73-s + 0.577·75-s + 0.450·79-s − 1/9·81-s + 2.63·83-s + 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.447793301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.447793301\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.59523025583029202558354503309, −10.47943702209173908529826735682, −10.16106844341492498816097585141, −9.458419761601227879363089501348, −9.116124051667141690885045631184, −8.702602132672200820601415364440, −8.443935749541386460275208542013, −7.978727741737425187783778722772, −7.19367553515283518122311309949, −6.80379843620544208445214810698, −6.55876832554089240168348208687, −6.18573546364636573254799723821, −5.02499277950116262149259163091, −4.94150342410443692419104271924, −4.40520587979508663278659154208, −3.63324610500658274442201888781, −3.15071750733572713514677841518, −2.58247034454246733586331312893, −1.77118414331097806789375930088, −0.914534402024743180882876203600,
0.914534402024743180882876203600, 1.77118414331097806789375930088, 2.58247034454246733586331312893, 3.15071750733572713514677841518, 3.63324610500658274442201888781, 4.40520587979508663278659154208, 4.94150342410443692419104271924, 5.02499277950116262149259163091, 6.18573546364636573254799723821, 6.55876832554089240168348208687, 6.80379843620544208445214810698, 7.19367553515283518122311309949, 7.978727741737425187783778722772, 8.443935749541386460275208542013, 8.702602132672200820601415364440, 9.116124051667141690885045631184, 9.458419761601227879363089501348, 10.16106844341492498816097585141, 10.47943702209173908529826735682, 10.59523025583029202558354503309
