| L(s) = 1 | + 3-s − 7-s − 2·9-s + 2·11-s + 13-s + 6·19-s − 21-s + 23-s − 5·27-s + 29-s − 31-s + 2·33-s + 6·37-s + 39-s + 3·41-s − 3·47-s + 49-s − 6·53-s + 6·57-s − 8·59-s − 10·61-s + 2·63-s − 2·67-s + 69-s − 5·71-s + 7·73-s − 2·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s + 1.37·19-s − 0.218·21-s + 0.208·23-s − 0.962·27-s + 0.185·29-s − 0.179·31-s + 0.348·33-s + 0.986·37-s + 0.160·39-s + 0.468·41-s − 0.437·47-s + 1/7·49-s − 0.824·53-s + 0.794·57-s − 1.04·59-s − 1.28·61-s + 0.251·63-s − 0.244·67-s + 0.120·69-s − 0.593·71-s + 0.819·73-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.28927058214746, −14.13794232011178, −13.52719869286368, −13.11166969018195, −12.45874733050422, −11.98195016603039, −11.37998417771370, −11.11964367823845, −10.38427004723713, −9.622957581229150, −9.467768839088795, −8.861328161546193, −8.405914240792761, −7.695070697301613, −7.405882320614445, −6.633614041751502, −6.024405451704865, −5.692684091191265, −4.870397116429006, −4.292164889518952, −3.516892780479387, −3.107705501348945, −2.611242707494526, −1.679186209265867, −1.021681413555125, 0,
1.021681413555125, 1.679186209265867, 2.611242707494526, 3.107705501348945, 3.516892780479387, 4.292164889518952, 4.870397116429006, 5.692684091191265, 6.024405451704865, 6.633614041751502, 7.405882320614445, 7.695070697301613, 8.405914240792761, 8.861328161546193, 9.467768839088795, 9.622957581229150, 10.38427004723713, 11.11964367823845, 11.37998417771370, 11.98195016603039, 12.45874733050422, 13.11166969018195, 13.52719869286368, 14.13794232011178, 14.28927058214746
