L(s) = 1 | − 2·3-s + 4·5-s − 7-s + 9-s − 2·11-s − 4·13-s − 8·15-s + 2·17-s + 6·19-s + 2·21-s + 11·25-s + 4·27-s + 8·29-s − 8·31-s + 4·33-s − 4·35-s + 8·37-s + 8·39-s − 10·41-s − 2·43-s + 4·45-s + 8·47-s + 49-s − 4·51-s − 8·55-s − 12·57-s + 10·59-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 2.06·15-s + 0.485·17-s + 1.37·19-s + 0.436·21-s + 11/5·25-s + 0.769·27-s + 1.48·29-s − 1.43·31-s + 0.696·33-s − 0.676·35-s + 1.31·37-s + 1.28·39-s − 1.56·41-s − 0.304·43-s + 0.596·45-s + 1.16·47-s + 1/7·49-s − 0.560·51-s − 1.07·55-s − 1.58·57-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.410289987\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.410289987\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.650714546944801590911855994156, −8.678643686317649477718644247599, −7.42540308558922446084693404501, −6.69412652552755616214187931000, −5.88696576012629669832053264825, −5.34778222172203362180971206312, −4.86921700397410511963661972291, −3.10642533389516736373234129624, −2.21810442891004784054451882023, −0.863854104924454366506812175947,
0.863854104924454366506812175947, 2.21810442891004784054451882023, 3.10642533389516736373234129624, 4.86921700397410511963661972291, 5.34778222172203362180971206312, 5.88696576012629669832053264825, 6.69412652552755616214187931000, 7.42540308558922446084693404501, 8.678643686317649477718644247599, 9.650714546944801590911855994156