| L(s) = 1 | + 2.47·2-s + 1.05·3-s + 4.11·4-s + 2.60·6-s + 3.75·7-s + 5.21·8-s − 1.88·9-s − 4.39·11-s + 4.33·12-s − 2.86·13-s + 9.27·14-s + 4.67·16-s + 7.95·17-s − 4.67·18-s + 2.24·19-s + 3.95·21-s − 10.8·22-s + 8.91·23-s + 5.49·24-s − 7.09·26-s − 5.15·27-s + 15.4·28-s + 8.73·29-s + 3.99·31-s + 1.12·32-s − 4.63·33-s + 19.6·34-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 0.608·3-s + 2.05·4-s + 1.06·6-s + 1.41·7-s + 1.84·8-s − 0.629·9-s − 1.32·11-s + 1.25·12-s − 0.795·13-s + 2.47·14-s + 1.16·16-s + 1.92·17-s − 1.10·18-s + 0.516·19-s + 0.862·21-s − 2.31·22-s + 1.85·23-s + 1.12·24-s − 1.39·26-s − 0.991·27-s + 2.91·28-s + 1.62·29-s + 0.717·31-s + 0.199·32-s − 0.806·33-s + 3.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.385352007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.385352007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 3 | \( 1 - 1.05T + 3T^{2} \) |
| 7 | \( 1 - 3.75T + 7T^{2} \) |
| 11 | \( 1 + 4.39T + 11T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 - 8.91T + 23T^{2} \) |
| 29 | \( 1 - 8.73T + 29T^{2} \) |
| 31 | \( 1 - 3.99T + 31T^{2} \) |
| 37 | \( 1 - 2.49T + 37T^{2} \) |
| 41 | \( 1 + 6.91T + 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 + 5.01T + 47T^{2} \) |
| 53 | \( 1 + 7.55T + 53T^{2} \) |
| 59 | \( 1 - 3.02T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 - 1.63T + 67T^{2} \) |
| 71 | \( 1 - 3.87T + 71T^{2} \) |
| 73 | \( 1 + 6.49T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 6.37T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−8.105875792074873135357186304994, −7.33477572015455197080024726204, −6.53839699947104118214103389554, −5.31386027206412723635102688611, −5.23506123553532549961842925073, −4.74436037731038017728948855885, −3.52225159941840833702117964067, −2.90064469920881978399328301706, −2.44057493644430602120587693012, −1.23956830718248515210738628013,
1.23956830718248515210738628013, 2.44057493644430602120587693012, 2.90064469920881978399328301706, 3.52225159941840833702117964067, 4.74436037731038017728948855885, 5.23506123553532549961842925073, 5.31386027206412723635102688611, 6.53839699947104118214103389554, 7.33477572015455197080024726204, 8.105875792074873135357186304994
