L(s) = 1 | + 3-s + 2.82·7-s + 9-s − 6.46·11-s − 13-s + 3.49·17-s − 2.21·19-s + 2.82·21-s − 7.14·23-s + 27-s + 4.82·29-s − 9.65·31-s − 6.46·33-s + 6.93·37-s − 39-s − 0.148·41-s + 3.03·43-s + 6.79·47-s + 1.00·49-s + 3.49·51-s + 1.70·53-s − 2.21·57-s − 6.46·59-s + 2.66·61-s + 2.82·63-s − 7.70·67-s − 7.14·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.06·7-s + 0.333·9-s − 1.95·11-s − 0.277·13-s + 0.847·17-s − 0.507·19-s + 0.617·21-s − 1.49·23-s + 0.192·27-s + 0.896·29-s − 1.73·31-s − 1.12·33-s + 1.14·37-s − 0.160·39-s − 0.0232·41-s + 0.463·43-s + 0.990·47-s + 0.142·49-s + 0.489·51-s + 0.233·53-s − 0.292·57-s − 0.842·59-s + 0.341·61-s + 0.356·63-s − 0.941·67-s − 0.860·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 6.46T + 11T^{2} \) |
| 17 | \( 1 - 3.49T + 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 + 7.14T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 9.65T + 31T^{2} \) |
| 37 | \( 1 - 6.93T + 37T^{2} \) |
| 41 | \( 1 + 0.148T + 41T^{2} \) |
| 43 | \( 1 - 3.03T + 43T^{2} \) |
| 47 | \( 1 - 6.79T + 47T^{2} \) |
| 53 | \( 1 - 1.70T + 53T^{2} \) |
| 59 | \( 1 + 6.46T + 59T^{2} \) |
| 61 | \( 1 - 2.66T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 5.76T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 2.27T + 83T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 + 8.11T + 97T^{2} \) |
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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
−7.72790062672760608738542524216, −7.13671775914136178670613300991, −5.87493998444514504985168377333, −5.45848866272016351322494616666, −4.62869307386204083014390507174, −4.01451061242531986994747150452, −2.89243853343559977716115035950, −2.34294414123532789363412968147, −1.46401820076699683352339139584, 0,
1.46401820076699683352339139584, 2.34294414123532789363412968147, 2.89243853343559977716115035950, 4.01451061242531986994747150452, 4.62869307386204083014390507174, 5.45848866272016351322494616666, 5.87493998444514504985168377333, 7.13671775914136178670613300991, 7.72790062672760608738542524216