| L(s) = 1 | − 3-s − 5-s + 0.681·7-s + 9-s + 3.18·11-s + 13-s + 15-s + 3.69·17-s + 3.87·19-s − 0.681·21-s + 6.04·23-s + 25-s − 27-s + 4.85·29-s − 4·31-s − 3.18·33-s − 0.681·35-s + 5.18·37-s − 39-s + 1.18·41-s + 6.37·43-s − 45-s − 10.7·47-s − 6.53·49-s − 3.69·51-s − 7.56·53-s − 3.18·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.257·7-s + 0.333·9-s + 0.961·11-s + 0.277·13-s + 0.258·15-s + 0.896·17-s + 0.888·19-s − 0.148·21-s + 1.26·23-s + 0.200·25-s − 0.192·27-s + 0.901·29-s − 0.718·31-s − 0.555·33-s − 0.115·35-s + 0.853·37-s − 0.160·39-s + 0.185·41-s + 0.972·43-s − 0.149·45-s − 1.56·47-s − 0.933·49-s − 0.517·51-s − 1.03·53-s − 0.430·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.879999407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.879999407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 0.681T + 7T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 - 3.87T + 19T^{2} \) |
| 23 | \( 1 - 6.04T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 - 6.37T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 - 6.55T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 1.82T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 2.72T + 83T^{2} \) |
| 89 | \( 1 + 4.17T + 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
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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
−7.956014052703332188345336757181, −7.30724760458632725728739161778, −6.64366653482579490182595855018, −5.94424414758466026745882862043, −5.13556563300578095314200385374, −4.52982785267199825028336350513, −3.62872994042366668895248465251, −2.96327305345754045850621035616, −1.52064480685058520221892285901, −0.817351016587291597509076379066,
0.817351016587291597509076379066, 1.52064480685058520221892285901, 2.96327305345754045850621035616, 3.62872994042366668895248465251, 4.52982785267199825028336350513, 5.13556563300578095314200385374, 5.94424414758466026745882862043, 6.64366653482579490182595855018, 7.30724760458632725728739161778, 7.956014052703332188345336757181
