L(s) = 1 | + 1.35·2-s − 0.160·4-s − 2.74·5-s − 0.283·7-s − 2.93·8-s − 3.71·10-s + 4.12·11-s + 0.0271·13-s − 0.384·14-s − 3.65·16-s + 1.28·17-s − 5.72·19-s + 0.440·20-s + 5.59·22-s + 5.97·23-s + 2.51·25-s + 0.0368·26-s + 0.0455·28-s + 2.55·29-s − 2.02·31-s + 0.905·32-s + 1.74·34-s + 0.777·35-s + 2.42·37-s − 7.76·38-s + 8.03·40-s + 11.0·41-s + ⋯ |
L(s) = 1 | + 0.959·2-s − 0.0802·4-s − 1.22·5-s − 0.107·7-s − 1.03·8-s − 1.17·10-s + 1.24·11-s + 0.00754·13-s − 0.102·14-s − 0.913·16-s + 0.312·17-s − 1.31·19-s + 0.0984·20-s + 1.19·22-s + 1.24·23-s + 0.503·25-s + 0.00723·26-s + 0.00860·28-s + 0.474·29-s − 0.364·31-s + 0.160·32-s + 0.299·34-s + 0.131·35-s + 0.399·37-s − 1.26·38-s + 1.27·40-s + 1.72·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.850353682\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.850353682\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 - 1.35T + 2T^{2} \) |
| 5 | \( 1 + 2.74T + 5T^{2} \) |
| 7 | \( 1 + 0.283T + 7T^{2} \) |
| 11 | \( 1 - 4.12T + 11T^{2} \) |
| 13 | \( 1 - 0.0271T + 13T^{2} \) |
| 17 | \( 1 - 1.28T + 17T^{2} \) |
| 19 | \( 1 + 5.72T + 19T^{2} \) |
| 23 | \( 1 - 5.97T + 23T^{2} \) |
| 29 | \( 1 - 2.55T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 4.54T + 47T^{2} \) |
| 53 | \( 1 - 9.30T + 53T^{2} \) |
| 59 | \( 1 - 9.94T + 59T^{2} \) |
| 61 | \( 1 - 8.17T + 61T^{2} \) |
| 67 | \( 1 - 4.40T + 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 6.69T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 + 0.746T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.936915464536973320181212876415, −8.391200308786206227372542552620, −7.38174040045307799174717263422, −6.61307967051733956806723981411, −5.86564263801886190400259397354, −4.80283107666351012706813400405, −4.08514881967396252029764307237, −3.67773855518284955872201957363, −2.58525179049367731892609542942, −0.77466463474000457330126862464,
0.77466463474000457330126862464, 2.58525179049367731892609542942, 3.67773855518284955872201957363, 4.08514881967396252029764307237, 4.80283107666351012706813400405, 5.86564263801886190400259397354, 6.61307967051733956806723981411, 7.38174040045307799174717263422, 8.391200308786206227372542552620, 8.936915464536973320181212876415