| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 1.56·7-s − 8-s + 9-s − 1.56·11-s + 12-s − 2·13-s − 1.56·14-s + 16-s − 5.12·17-s − 18-s + 4.68·19-s + 1.56·21-s + 1.56·22-s − 5.56·23-s − 24-s + 2·26-s + 27-s + 1.56·28-s − 1.12·29-s + 31-s − 32-s − 1.56·33-s + 5.12·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.590·7-s − 0.353·8-s + 0.333·9-s − 0.470·11-s + 0.288·12-s − 0.554·13-s − 0.417·14-s + 0.250·16-s − 1.24·17-s − 0.235·18-s + 1.07·19-s + 0.340·21-s + 0.332·22-s − 1.15·23-s − 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.295·28-s − 0.208·29-s + 0.179·31-s − 0.176·32-s − 0.271·33-s + 0.878·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 7.80T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 4.87T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 9.36T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 + 9.80T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 + 1.31T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.88072440267685758887714933776, −7.54559110576752732823903880263, −6.73450002133309116673581181493, −5.83717460423254574718919824469, −4.94169611714577999233259931418, −4.17419341567563567735975901158, −3.08888588453098945707356055268, −2.29109399853502836457665204763, −1.49208978547511272503541441005, 0,
1.49208978547511272503541441005, 2.29109399853502836457665204763, 3.08888588453098945707356055268, 4.17419341567563567735975901158, 4.94169611714577999233259931418, 5.83717460423254574718919824469, 6.73450002133309116673581181493, 7.54559110576752732823903880263, 7.88072440267685758887714933776
