L(s) = 1 | − 1.76·3-s + 5-s − 1.99·7-s + 0.105·9-s + 6.19·11-s + 4.77·13-s − 1.76·15-s + 1.12·17-s + 1.75·19-s + 3.52·21-s + 5.41·23-s + 25-s + 5.10·27-s + 3.31·29-s + 7.84·31-s − 10.9·33-s − 1.99·35-s − 9.11·37-s − 8.40·39-s + 4.18·41-s + 0.462·43-s + 0.105·45-s − 2.98·47-s − 3.00·49-s − 1.98·51-s − 7.03·53-s + 6.19·55-s + ⋯ |
L(s) = 1 | − 1.01·3-s + 0.447·5-s − 0.755·7-s + 0.0351·9-s + 1.86·11-s + 1.32·13-s − 0.454·15-s + 0.272·17-s + 0.402·19-s + 0.768·21-s + 1.12·23-s + 0.200·25-s + 0.981·27-s + 0.614·29-s + 1.40·31-s − 1.89·33-s − 0.337·35-s − 1.49·37-s − 1.34·39-s + 0.653·41-s + 0.0705·43-s + 0.0156·45-s − 0.434·47-s − 0.429·49-s − 0.277·51-s − 0.965·53-s + 0.834·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.753236882\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.753236882\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
good | 3 | \( 1 + 1.76T + 3T^{2} \) |
| 7 | \( 1 + 1.99T + 7T^{2} \) |
| 11 | \( 1 - 6.19T + 11T^{2} \) |
| 13 | \( 1 - 4.77T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 - 3.31T + 29T^{2} \) |
| 31 | \( 1 - 7.84T + 31T^{2} \) |
| 37 | \( 1 + 9.11T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 - 0.462T + 43T^{2} \) |
| 47 | \( 1 + 2.98T + 47T^{2} \) |
| 53 | \( 1 + 7.03T + 53T^{2} \) |
| 59 | \( 1 + 1.24T + 59T^{2} \) |
| 61 | \( 1 - 1.09T + 61T^{2} \) |
| 67 | \( 1 + 1.00T + 67T^{2} \) |
| 71 | \( 1 - 3.09T + 71T^{2} \) |
| 73 | \( 1 - 2.08T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 2.22T + 83T^{2} \) |
| 89 | \( 1 + 5.23T + 89T^{2} \) |
| 97 | \( 1 - 0.579T + 97T^{2} \) |
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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
−8.206476833666349945074077367557, −6.90742554964742044716401681016, −6.50810284498120725163366404903, −6.14689002481200702531771695510, −5.35356835339489017916988419525, −4.54308683102105336648494922923, −3.59999307468182344374931618534, −2.98173328093055795095118768328, −1.46103153189072953697171224939, −0.825819776520468999183663427840,
0.825819776520468999183663427840, 1.46103153189072953697171224939, 2.98173328093055795095118768328, 3.59999307468182344374931618534, 4.54308683102105336648494922923, 5.35356835339489017916988419525, 6.14689002481200702531771695510, 6.50810284498120725163366404903, 6.90742554964742044716401681016, 8.206476833666349945074077367557