| L(s) = 1 | − 3-s − 3.01·5-s − 1.84·7-s + 9-s − 0.0702·11-s + 1.68·13-s + 3.01·15-s − 2.17·17-s − 2.90·19-s + 1.84·21-s + 1.67·23-s + 4.09·25-s − 27-s − 4.24·29-s − 4.28·31-s + 0.0702·33-s + 5.56·35-s − 10.8·37-s − 1.68·39-s − 7.20·41-s − 4.33·43-s − 3.01·45-s + 11.5·47-s − 3.58·49-s + 2.17·51-s + 3.88·53-s + 0.211·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.698·7-s + 0.333·9-s − 0.0211·11-s + 0.468·13-s + 0.778·15-s − 0.527·17-s − 0.667·19-s + 0.402·21-s + 0.348·23-s + 0.818·25-s − 0.192·27-s − 0.787·29-s − 0.770·31-s + 0.0122·33-s + 0.941·35-s − 1.78·37-s − 0.270·39-s − 1.12·41-s − 0.661·43-s − 0.449·45-s + 1.69·47-s − 0.512·49-s + 0.304·51-s + 0.533·53-s + 0.0285·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4911758285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4911758285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 3.01T + 5T^{2} \) |
| 7 | \( 1 + 1.84T + 7T^{2} \) |
| 11 | \( 1 + 0.0702T + 11T^{2} \) |
| 13 | \( 1 - 1.68T + 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 + 2.90T + 19T^{2} \) |
| 23 | \( 1 - 1.67T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 4.28T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 7.20T + 41T^{2} \) |
| 43 | \( 1 + 4.33T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 3.88T + 53T^{2} \) |
| 59 | \( 1 - 5.40T + 59T^{2} \) |
| 61 | \( 1 - 2.01T + 61T^{2} \) |
| 67 | \( 1 + 6.98T + 67T^{2} \) |
| 71 | \( 1 - 2.15T + 71T^{2} \) |
| 73 | \( 1 + 8.46T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 9.35T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.651225490213709003787593443510, −7.48794258349876604763315823330, −7.08026548111216092188330833379, −6.31917697519767565345084348129, −5.48299002498714623777039742387, −4.59105994844044366997630142356, −3.80956486970616755247145068805, −3.27399015798948725841846118670, −1.87686017109153243897704234037, −0.40012813835925092910818212629,
0.40012813835925092910818212629, 1.87686017109153243897704234037, 3.27399015798948725841846118670, 3.80956486970616755247145068805, 4.59105994844044366997630142356, 5.48299002498714623777039742387, 6.31917697519767565345084348129, 7.08026548111216092188330833379, 7.48794258349876604763315823330, 8.651225490213709003787593443510
