| L(s) = 1 | − 0.414·3-s − 5-s − 2.82·9-s − 3.82·11-s + 3.58·13-s + 0.414·15-s + 6.41·17-s + 3.65·19-s + 0.585·23-s + 25-s + 2.41·27-s + 6.65·29-s + 4.58·31-s + 1.58·33-s − 3.41·37-s − 1.48·39-s + 0.585·41-s + 11.6·43-s + 2.82·45-s − 8.89·47-s − 2.65·51-s − 3.75·53-s + 3.82·55-s − 1.51·57-s + 3.41·59-s + 5.17·61-s − 3.58·65-s + ⋯ |
| L(s) = 1 | − 0.239·3-s − 0.447·5-s − 0.942·9-s − 1.15·11-s + 0.994·13-s + 0.106·15-s + 1.55·17-s + 0.838·19-s + 0.122·23-s + 0.200·25-s + 0.464·27-s + 1.23·29-s + 0.823·31-s + 0.276·33-s − 0.561·37-s − 0.237·39-s + 0.0914·41-s + 1.77·43-s + 0.421·45-s − 1.29·47-s − 0.372·51-s − 0.516·53-s + 0.516·55-s − 0.200·57-s + 0.444·59-s + 0.662·61-s − 0.444·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.239220942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.239220942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 - 6.41T + 17T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 23 | \( 1 - 0.585T + 23T^{2} \) |
| 29 | \( 1 - 6.65T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 - 0.585T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 8.89T + 47T^{2} \) |
| 53 | \( 1 + 3.75T + 53T^{2} \) |
| 59 | \( 1 - 3.41T + 59T^{2} \) |
| 61 | \( 1 - 5.17T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−10.18181289337866264006377234543, −9.104550421850935656851454936345, −8.125291161511137798438079282916, −7.77418011814709207573770323301, −6.49222971907221780121271830978, −5.61668618531821323409870696066, −4.92618062324354990304610105151, −3.52273962006566450914630672286, −2.77078799651627360286253031598, −0.894208462432729505905730397256,
0.894208462432729505905730397256, 2.77078799651627360286253031598, 3.52273962006566450914630672286, 4.92618062324354990304610105151, 5.61668618531821323409870696066, 6.49222971907221780121271830978, 7.77418011814709207573770323301, 8.125291161511137798438079282916, 9.104550421850935656851454936345, 10.18181289337866264006377234543
