L(s) = 1 | − 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 − 10·53-s + 14·59-s − 4·61-s + 3·67-s − 13·71-s − 16·73-s + 79-s + 9·81-s − 10·83-s − 10·89-s + 2·97-s + 15·99-s − 12·101-s − 2·103-s − 12·107-s + ⋯ |
L(s) = 1 | − 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 − 1.37·53-s + 1.82·59-s − 0.512·61-s + 0.366·67-s − 1.54·71-s − 1.87·73-s + 0.112·79-s + 81-s − 1.09·83-s − 1.05·89-s + 0.203·97-s + 1.50·99-s − 1.19·101-s − 0.197·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
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
−7.968196762134285462571255838165, −7.29138066151960322079895177090, −6.33369483233931101373536455234, −5.65706902447689129260739654779, −5.15615989958638804242802509519, −4.11582013407297080823933504962, −3.14932756834754017490579396539, −2.61071617189497754670299945635, −1.32078418947378566442652592258, 0,
1.32078418947378566442652592258, 2.61071617189497754670299945635, 3.14932756834754017490579396539, 4.11582013407297080823933504962, 5.15615989958638804242802509519, 5.65706902447689129260739654779, 6.33369483233931101373536455234, 7.29138066151960322079895177090, 7.968196762134285462571255838165