| L(s) = 1 | + 7-s − 3·9-s + 5·11-s + 6·13-s − 4·17-s + 6·19-s + 3·23-s − 3·29-s − 2·31-s − 7·37-s − 4·41-s − 7·43-s − 2·47-s + 49-s + 10·53-s + 14·59-s + 4·61-s − 3·63-s + 3·67-s + 13·71-s − 16·73-s + 5·77-s − 79-s + 9·81-s + 10·83-s + 10·89-s + 6·91-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 1.50·11-s + 1.66·13-s − 0.970·17-s + 1.37·19-s + 0.625·23-s − 0.557·29-s − 0.359·31-s − 1.15·37-s − 0.624·41-s − 1.06·43-s − 0.291·47-s + 1/7·49-s + 1.37·53-s + 1.82·59-s + 0.512·61-s − 0.377·63-s + 0.366·67-s + 1.54·71-s − 1.87·73-s + 0.569·77-s − 0.112·79-s + 81-s + 1.09·83-s + 1.05·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.118632288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.118632288\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.771331736120762292476444340292, −8.333444845853473479186197797355, −7.13812072569312645525222426140, −6.55099287395857952449351544384, −5.73484540211776229624490804870, −5.01568922144647406561931121310, −3.79724663650266550279587469144, −3.38005594991122115399644240303, −1.96482026466734574438362911543, −0.957036473863583182042762876360,
0.957036473863583182042762876360, 1.96482026466734574438362911543, 3.38005594991122115399644240303, 3.79724663650266550279587469144, 5.01568922144647406561931121310, 5.73484540211776229624490804870, 6.55099287395857952449351544384, 7.13812072569312645525222426140, 8.333444845853473479186197797355, 8.771331736120762292476444340292
