| L(s) = 1 | + 3-s + 9-s − 11-s + 2·13-s + 6·17-s + 27-s − 2·29-s − 33-s + 2·37-s + 2·39-s + 6·41-s + 4·43-s + 8·47-s + 6·51-s + 6·53-s − 10·61-s − 8·67-s + 16·71-s − 10·73-s + 8·79-s + 81-s + 12·83-s − 2·87-s + 10·89-s − 10·97-s − 99-s + 101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 1.45·17-s + 0.192·27-s − 0.371·29-s − 0.174·33-s + 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 0.840·51-s + 0.824·53-s − 1.28·61-s − 0.977·67-s + 1.89·71-s − 1.17·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-s − 0.214·87-s + 1.05·89-s − 1.01·97-s − 0.100·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.490180750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.490180750\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−12.51317978797883, −12.21386590371659, −11.91036150507344, −11.04447622743681, −10.84485695161125, −10.36588030852816, −9.783991466622322, −9.431461103413649, −8.966507671920514, −8.487324502027448, −7.949358291258143, −7.545961928376250, −7.301590213029911, −6.519592395313670, −5.994993213470921, −5.637298793401184, −5.070738930965316, −4.460250785850938, −3.886535840482065, −3.507811411321524, −2.902396971534078, −2.430939820991115, −1.771120680144535, −1.089330381928203, −0.5935587313668583,
0.5935587313668583, 1.089330381928203, 1.771120680144535, 2.430939820991115, 2.902396971534078, 3.507811411321524, 3.886535840482065, 4.460250785850938, 5.070738930965316, 5.637298793401184, 5.994993213470921, 6.519592395313670, 7.301590213029911, 7.545961928376250, 7.949358291258143, 8.487324502027448, 8.966507671920514, 9.431461103413649, 9.783991466622322, 10.36588030852816, 10.84485695161125, 11.04447622743681, 11.91036150507344, 12.21386590371659, 12.51317978797883
