| L(s) = 1 | + 2-s + 1.24·3-s + 4-s + 1.24·6-s − 2.80·7-s + 8-s − 1.44·9-s + 0.554·11-s + 1.24·12-s − 2.80·14-s + 16-s + 3.24·17-s − 1.44·18-s − 3.24·19-s − 3.49·21-s + 0.554·22-s − 4.04·23-s + 1.24·24-s − 5.54·27-s − 2.80·28-s + 0.384·29-s + 3.13·31-s + 32-s + 0.692·33-s + 3.24·34-s − 1.44·36-s + 11.3·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.719·3-s + 0.5·4-s + 0.509·6-s − 1.05·7-s + 0.353·8-s − 0.481·9-s + 0.167·11-s + 0.359·12-s − 0.748·14-s + 0.250·16-s + 0.787·17-s − 0.340·18-s − 0.744·19-s − 0.762·21-s + 0.118·22-s − 0.844·23-s + 0.254·24-s − 1.06·27-s − 0.529·28-s + 0.0713·29-s + 0.563·31-s + 0.176·32-s + 0.120·33-s + 0.556·34-s − 0.240·36-s + 1.86·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.24T + 3T^{2} \) |
| 7 | \( 1 + 2.80T + 7T^{2} \) |
| 11 | \( 1 - 0.554T + 11T^{2} \) |
| 17 | \( 1 - 3.24T + 17T^{2} \) |
| 19 | \( 1 + 3.24T + 19T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 - 0.384T + 29T^{2} \) |
| 31 | \( 1 - 3.13T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 1.96T + 41T^{2} \) |
| 43 | \( 1 + 8.32T + 43T^{2} \) |
| 47 | \( 1 - 3.78T + 47T^{2} \) |
| 53 | \( 1 + 7.92T + 53T^{2} \) |
| 59 | \( 1 - 9.04T + 59T^{2} \) |
| 61 | \( 1 + 9.04T + 61T^{2} \) |
| 67 | \( 1 + 0.396T + 67T^{2} \) |
| 71 | \( 1 - 1.54T + 71T^{2} \) |
| 73 | \( 1 + 4.03T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.42090737249560494708028799450, −6.59696673486873133179202618124, −6.06202374907489289192150446468, −5.49614772187334508965510931961, −4.40702078242099648328008597018, −3.83173008935359370463452125292, −2.99835058288355761939885423522, −2.64335417685094178374750703090, −1.51559806645654557012172958144, 0,
1.51559806645654557012172958144, 2.64335417685094178374750703090, 2.99835058288355761939885423522, 3.83173008935359370463452125292, 4.40702078242099648328008597018, 5.49614772187334508965510931961, 6.06202374907489289192150446468, 6.59696673486873133179202618124, 7.42090737249560494708028799450
