| L(s) = 1 | + 0.878·3-s − 1.66·5-s − 1.34·7-s − 2.22·9-s − 11-s + 0.817·13-s − 1.46·15-s + 3.28·17-s + 19-s − 1.18·21-s − 3.16·23-s − 2.22·25-s − 4.59·27-s − 5.77·29-s + 8.86·31-s − 0.878·33-s + 2.24·35-s + 11.8·37-s + 0.718·39-s + 4.14·41-s + 9.25·43-s + 3.70·45-s + 1.15·47-s − 5.17·49-s + 2.88·51-s − 9.30·53-s + 1.66·55-s + ⋯ |
| L(s) = 1 | + 0.507·3-s − 0.744·5-s − 0.510·7-s − 0.742·9-s − 0.301·11-s + 0.226·13-s − 0.377·15-s + 0.796·17-s + 0.229·19-s − 0.258·21-s − 0.659·23-s − 0.445·25-s − 0.884·27-s − 1.07·29-s + 1.59·31-s − 0.152·33-s + 0.379·35-s + 1.94·37-s + 0.115·39-s + 0.647·41-s + 1.41·43-s + 0.552·45-s + 0.168·47-s − 0.739·49-s + 0.404·51-s − 1.27·53-s + 0.224·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.434224294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.434224294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.878T + 3T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 + 1.34T + 7T^{2} \) |
| 13 | \( 1 - 0.817T + 13T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + 5.77T + 29T^{2} \) |
| 31 | \( 1 - 8.86T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 - 4.14T + 41T^{2} \) |
| 43 | \( 1 - 9.25T + 43T^{2} \) |
| 47 | \( 1 - 1.15T + 47T^{2} \) |
| 53 | \( 1 + 9.30T + 53T^{2} \) |
| 59 | \( 1 - 9.09T + 59T^{2} \) |
| 61 | \( 1 + 8.45T + 61T^{2} \) |
| 67 | \( 1 - 1.83T + 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 - 7.79T + 73T^{2} \) |
| 79 | \( 1 + 0.830T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 3.06T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.427690408269553105665916220920, −7.891991952911506350516134411806, −7.46952900738011301705035036596, −6.20480846821070825959431210219, −5.80587340758350088016884898555, −4.64064840970555181160905471473, −3.77341782049303507541839090357, −3.11490106310496493211483630729, −2.26268208172427283803297899565, −0.67372576599023211874938158282,
0.67372576599023211874938158282, 2.26268208172427283803297899565, 3.11490106310496493211483630729, 3.77341782049303507541839090357, 4.64064840970555181160905471473, 5.80587340758350088016884898555, 6.20480846821070825959431210219, 7.46952900738011301705035036596, 7.891991952911506350516134411806, 8.427690408269553105665916220920
