| L(s) = 1 | + 2·3-s − 2·7-s + 9-s + 6·11-s + 4·13-s − 19-s − 4·21-s − 3·23-s − 4·27-s + 8·31-s + 12·33-s − 37-s + 8·39-s + 3·41-s + 43-s + 6·47-s − 3·49-s + 9·53-s − 2·57-s − 3·59-s + 14·61-s − 2·63-s + 4·67-s − 6·69-s − 6·71-s − 11·73-s − 12·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.80·11-s + 1.10·13-s − 0.229·19-s − 0.872·21-s − 0.625·23-s − 0.769·27-s + 1.43·31-s + 2.08·33-s − 0.164·37-s + 1.28·39-s + 0.468·41-s + 0.152·43-s + 0.875·47-s − 3/7·49-s + 1.23·53-s − 0.264·57-s − 0.390·59-s + 1.79·61-s − 0.251·63-s + 0.488·67-s − 0.722·69-s − 0.712·71-s − 1.28·73-s − 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.058656608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.058656608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.695989539686053355825686203246, −7.991004020788604507695291405594, −7.01356727322076751258902240396, −6.36288828129844455987369462362, −5.79035280065025086351126524762, −4.29307313133350837372675703697, −3.79269438867354264172300053577, −3.10982568936962903235731259689, −2.11398944405819991558864161567, −1.01602265548218522018617154684,
1.01602265548218522018617154684, 2.11398944405819991558864161567, 3.10982568936962903235731259689, 3.79269438867354264172300053577, 4.29307313133350837372675703697, 5.79035280065025086351126524762, 6.36288828129844455987369462362, 7.01356727322076751258902240396, 7.991004020788604507695291405594, 8.695989539686053355825686203246
