| L(s) = 1 | − 2-s − 3-s + 4-s − 3.09·5-s + 6-s + 3.97·7-s − 8-s + 9-s + 3.09·10-s − 2.08·11-s − 12-s − 13-s − 3.97·14-s + 3.09·15-s + 16-s + 3.07·17-s − 18-s − 6.35·19-s − 3.09·20-s − 3.97·21-s + 2.08·22-s + 0.753·23-s + 24-s + 4.57·25-s + 26-s − 27-s + 3.97·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.38·5-s + 0.408·6-s + 1.50·7-s − 0.353·8-s + 0.333·9-s + 0.978·10-s − 0.629·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 0.799·15-s + 0.250·16-s + 0.744·17-s − 0.235·18-s − 1.45·19-s − 0.692·20-s − 0.867·21-s + 0.445·22-s + 0.157·23-s + 0.204·24-s + 0.915·25-s + 0.196·26-s − 0.192·27-s + 0.750·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6706260245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6706260245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 3.09T + 5T^{2} \) |
| 7 | \( 1 - 3.97T + 7T^{2} \) |
| 11 | \( 1 + 2.08T + 11T^{2} \) |
| 17 | \( 1 - 3.07T + 17T^{2} \) |
| 19 | \( 1 + 6.35T + 19T^{2} \) |
| 23 | \( 1 - 0.753T + 23T^{2} \) |
| 29 | \( 1 + 7.73T + 29T^{2} \) |
| 31 | \( 1 - 5.41T + 31T^{2} \) |
| 37 | \( 1 - 6.26T + 37T^{2} \) |
| 41 | \( 1 + 3.19T + 41T^{2} \) |
| 43 | \( 1 - 4.21T + 43T^{2} \) |
| 47 | \( 1 + 9.09T + 47T^{2} \) |
| 53 | \( 1 - 1.78T + 53T^{2} \) |
| 59 | \( 1 - 3.06T + 59T^{2} \) |
| 61 | \( 1 + 7.26T + 61T^{2} \) |
| 67 | \( 1 + 4.59T + 67T^{2} \) |
| 71 | \( 1 - 6.81T + 71T^{2} \) |
| 73 | \( 1 + 1.90T + 73T^{2} \) |
| 79 | \( 1 - 3.61T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.87187992761329509154911037525, −7.48654221722557826290825800629, −6.62549010413530464787743165729, −5.75426386938774728741740912993, −4.92934811108068414213024729643, −4.41748950961285287882128432092, −3.60924200479765694240213647647, −2.46876311475653908375695291857, −1.54619037512751622868281149254, −0.47841399512480290524082160846,
0.47841399512480290524082160846, 1.54619037512751622868281149254, 2.46876311475653908375695291857, 3.60924200479765694240213647647, 4.41748950961285287882128432092, 4.92934811108068414213024729643, 5.75426386938774728741740912993, 6.62549010413530464787743165729, 7.48654221722557826290825800629, 7.87187992761329509154911037525
