L(s) = 1 | − 2-s + 1.24·3-s + 4-s − 1.24·6-s − 8-s + 0.554·9-s + 1.80·11-s + 1.24·12-s + 16-s − 1.80·17-s − 0.554·18-s + 0.445·19-s − 1.80·22-s − 1.24·24-s + 25-s − 0.554·27-s − 32-s + 2.24·33-s + 1.80·34-s + 0.554·36-s − 0.445·38-s + 0.445·41-s − 0.445·43-s + 1.80·44-s + 1.24·48-s + 49-s − 50-s + ⋯ |
L(s) = 1 | − 2-s + 1.24·3-s + 4-s − 1.24·6-s − 8-s + 0.554·9-s + 1.80·11-s + 1.24·12-s + 16-s − 1.80·17-s − 0.554·18-s + 0.445·19-s − 1.80·22-s − 1.24·24-s + 25-s − 0.554·27-s − 32-s + 2.24·33-s + 1.80·34-s + 0.554·36-s − 0.445·38-s + 0.445·41-s − 0.445·43-s + 1.80·44-s + 1.24·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.112929988\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.112929988\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 1.24T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.80T + T^{2} \) |
| 17 | \( 1 + 1.80T + T^{2} \) |
| 19 | \( 1 - 0.445T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.445T + T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.24T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.24T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.24T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.445T + T^{2} \) |
| 89 | \( 1 + 1.24T + T^{2} \) |
| 97 | \( 1 - 1.80T + T^{2} \) |
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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
−9.335006486566998005187032427874, −8.998161009629300678845992802965, −8.552374132242745331936134606213, −7.49620290506170427262453948143, −6.82103557523437074653198951012, −6.06743034957441844393410998142, −4.43102333568490005881517382781, −3.44407131966947999155178676608, −2.49146545336076355069241483812, −1.46715953389719651997465312122,
1.46715953389719651997465312122, 2.49146545336076355069241483812, 3.44407131966947999155178676608, 4.43102333568490005881517382781, 6.06743034957441844393410998142, 6.82103557523437074653198951012, 7.49620290506170427262453948143, 8.552374132242745331936134606213, 8.998161009629300678845992802965, 9.335006486566998005187032427874