L(s) = 1 | − 2-s + 3-s + 4-s − 3.46·5-s − 6-s − 8-s + 9-s + 3.46·10-s + 1.46·11-s + 12-s − 3.46·13-s − 3.46·15-s + 16-s − 0.535·17-s − 18-s − 19-s − 3.46·20-s − 1.46·22-s + 1.46·23-s − 24-s + 6.99·25-s + 3.46·26-s + 27-s − 6·29-s + 3.46·30-s + 6.92·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.54·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.09·10-s + 0.441·11-s + 0.288·12-s − 0.960·13-s − 0.894·15-s + 0.250·16-s − 0.129·17-s − 0.235·18-s − 0.229·19-s − 0.774·20-s − 0.312·22-s + 0.305·23-s − 0.204·24-s + 1.39·25-s + 0.679·26-s + 0.192·27-s − 1.11·29-s + 0.632·30-s + 1.24·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 0.535T + 17T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 - 2.53T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.81527249228281624034637417497, −7.37640818060576654741324481379, −6.73312677699122140700257445786, −5.75180339170024169886293020272, −4.49164835732825445009880234514, −4.14574308441539835543886498604, −3.11307018219718537892976293007, −2.45238323642217898084925885612, −1.14197829333841597024086891974, 0,
1.14197829333841597024086891974, 2.45238323642217898084925885612, 3.11307018219718537892976293007, 4.14574308441539835543886498604, 4.49164835732825445009880234514, 5.75180339170024169886293020272, 6.73312677699122140700257445786, 7.37640818060576654741324481379, 7.81527249228281624034637417497