| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s + 5·7-s + 9-s + 2·10-s + 2·11-s − 2·12-s − 6·13-s − 10·14-s + 15-s − 4·16-s − 2·18-s + 2·19-s − 2·20-s − 5·21-s − 4·22-s + 23-s + 25-s + 12·26-s − 27-s + 10·28-s + 29-s − 2·30-s + 5·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 1.88·7-s + 1/3·9-s + 0.632·10-s + 0.603·11-s − 0.577·12-s − 1.66·13-s − 2.67·14-s + 0.258·15-s − 16-s − 0.471·18-s + 0.458·19-s − 0.447·20-s − 1.09·21-s − 0.852·22-s + 0.208·23-s + 1/5·25-s + 2.35·26-s − 0.192·27-s + 1.88·28-s + 0.185·29-s − 0.365·30-s + 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99705 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99705 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.015477780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.015477780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.71561571096326, −13.39780831262481, −12.39478816142619, −12.03012157105492, −11.65040658235921, −11.25842471335840, −10.83166432317135, −10.28601946410999, −9.752623361010487, −9.377595446064628, −8.731561331377690, −8.192613757508338, −7.786630358631465, −7.508893226062053, −6.898283642786007, −6.417456417398038, −5.444959133101965, −5.012775214793285, −4.458441690624613, −4.229238674310422, −3.005462343560139, −2.294672729614163, −1.676773913590966, −1.099146571028098, −0.4862772367046520,
0.4862772367046520, 1.099146571028098, 1.676773913590966, 2.294672729614163, 3.005462343560139, 4.229238674310422, 4.458441690624613, 5.012775214793285, 5.444959133101965, 6.417456417398038, 6.898283642786007, 7.508893226062053, 7.786630358631465, 8.192613757508338, 8.731561331377690, 9.377595446064628, 9.752623361010487, 10.28601946410999, 10.83166432317135, 11.25842471335840, 11.65040658235921, 12.03012157105492, 12.39478816142619, 13.39780831262481, 13.71561571096326
