| L(s) = 1 | + 5-s + 4·11-s − 5·13-s + 5·17-s + 8·19-s + 4·23-s − 4·25-s − 3·29-s − 4·31-s + 3·37-s + 6·41-s + 4·43-s + 12·47-s − 7·49-s + 10·53-s + 4·55-s − 8·59-s − 5·61-s − 5·65-s + 8·67-s − 16·71-s − 5·73-s + 4·79-s − 4·83-s + 5·85-s − 3·89-s + 8·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s − 1.38·13-s + 1.21·17-s + 1.83·19-s + 0.834·23-s − 4/5·25-s − 0.557·29-s − 0.718·31-s + 0.493·37-s + 0.937·41-s + 0.609·43-s + 1.75·47-s − 49-s + 1.37·53-s + 0.539·55-s − 1.04·59-s − 0.640·61-s − 0.620·65-s + 0.977·67-s − 1.89·71-s − 0.585·73-s + 0.450·79-s − 0.439·83-s + 0.542·85-s − 0.317·89-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.712919003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.712919003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.41427784154370066336577907232, −9.441524620445526535175360937196, −9.276700222420769412340340101500, −7.65043355649184430630149859953, −7.22985671868851610688971612408, −5.91309717484246366542765564947, −5.21013759142456713415671940960, −3.93598076451906063518111572034, −2.76720787421132320912901821151, −1.26763827522611928708335465597,
1.26763827522611928708335465597, 2.76720787421132320912901821151, 3.93598076451906063518111572034, 5.21013759142456713415671940960, 5.91309717484246366542765564947, 7.22985671868851610688971612408, 7.65043355649184430630149859953, 9.276700222420769412340340101500, 9.441524620445526535175360937196, 10.41427784154370066336577907232
