| L(s) = 1 | − 2-s − 4-s − 2·7-s + 3·8-s + 4·13-s + 2·14-s − 16-s − 2·17-s − 2·23-s − 4·26-s + 2·28-s − 29-s + 4·31-s − 5·32-s + 2·34-s + 2·37-s − 10·41-s + 2·46-s − 12·47-s − 3·49-s − 4·52-s + 12·53-s − 6·56-s + 58-s − 4·59-s + 2·61-s − 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s + 1.10·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.417·23-s − 0.784·26-s + 0.377·28-s − 0.185·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.328·37-s − 1.56·41-s + 0.294·46-s − 1.75·47-s − 3/7·49-s − 0.554·52-s + 1.64·53-s − 0.801·56-s + 0.131·58-s − 0.520·59-s + 0.256·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.074852614018617119469164055026, −6.85177451755070890027036589411, −6.49813348955312197969181477913, −5.54374133889135242576506598653, −4.76833201393300904330534492770, −3.90201919560047035984163698931, −3.32208862701546345219374445048, −2.09338348333897711480297672337, −1.07712154987417689160552481476, 0,
1.07712154987417689160552481476, 2.09338348333897711480297672337, 3.32208862701546345219374445048, 3.90201919560047035984163698931, 4.76833201393300904330534492770, 5.54374133889135242576506598653, 6.49813348955312197969181477913, 6.85177451755070890027036589411, 8.074852614018617119469164055026
