L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 12-s − 2·13-s − 16-s + 2·17-s − 18-s − 4·19-s + 3·24-s + 2·26-s + 27-s + 2·29-s − 5·32-s − 2·34-s − 36-s + 10·37-s + 4·38-s − 2·39-s − 10·41-s + 4·43-s − 8·47-s − 48-s − 7·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.612·24-s + 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.883·32-s − 0.342·34-s − 1/6·36-s + 1.64·37-s + 0.648·38-s − 0.320·39-s − 1.56·41-s + 0.609·43-s − 1.16·47-s − 0.144·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.66467286781506385807539112274, −6.91978336503772440759710674717, −6.15292591976645848372017937772, −5.15835212184276367870069042340, −4.55792168261302849108519792726, −3.87454337592842405194837571943, −2.97354280975495339465015673245, −2.06760344491783741159565622727, −1.16312488453365808625765234194, 0,
1.16312488453365808625765234194, 2.06760344491783741159565622727, 2.97354280975495339465015673245, 3.87454337592842405194837571943, 4.55792168261302849108519792726, 5.15835212184276367870069042340, 6.15292591976645848372017937772, 6.91978336503772440759710674717, 7.66467286781506385807539112274