| L(s) = 1 | + 3·5-s + 7-s + 11-s + 5·13-s + 6·17-s − 19-s + 4·25-s − 9·29-s + 8·31-s + 3·35-s − 7·37-s + 6·41-s − 10·43-s + 3·47-s + 49-s + 6·53-s + 3·55-s − 9·59-s + 2·61-s + 15·65-s − 13·67-s + 6·71-s + 11·73-s + 77-s − 10·79-s − 6·83-s + 18·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s + 0.301·11-s + 1.38·13-s + 1.45·17-s − 0.229·19-s + 4/5·25-s − 1.67·29-s + 1.43·31-s + 0.507·35-s − 1.15·37-s + 0.937·41-s − 1.52·43-s + 0.437·47-s + 1/7·49-s + 0.824·53-s + 0.404·55-s − 1.17·59-s + 0.256·61-s + 1.86·65-s − 1.58·67-s + 0.712·71-s + 1.28·73-s + 0.113·77-s − 1.12·79-s − 0.658·83-s + 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2772 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.861611711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.861611711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.842594039086178326500591935376, −8.181420203004949073525973454498, −7.27016378051915738256112776875, −6.29319453144775499330311552858, −5.80848036100673777414578320532, −5.14087575750725443462854848300, −3.98314392755676846537550023468, −3.11127881507401601242628626937, −1.89516592560822348435158187789, −1.19268371912872473978099421960,
1.19268371912872473978099421960, 1.89516592560822348435158187789, 3.11127881507401601242628626937, 3.98314392755676846537550023468, 5.14087575750725443462854848300, 5.80848036100673777414578320532, 6.29319453144775499330311552858, 7.27016378051915738256112776875, 8.181420203004949073525973454498, 8.842594039086178326500591935376
