| L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s − 6·11-s + 13-s + 16-s − 3·17-s − 2·19-s + 3·20-s − 6·22-s − 6·23-s + 4·25-s + 26-s − 9·29-s + 10·31-s + 32-s − 3·34-s − 7·37-s − 2·38-s + 3·40-s − 6·41-s − 4·43-s − 6·44-s − 6·46-s − 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s − 1.80·11-s + 0.277·13-s + 1/4·16-s − 0.727·17-s − 0.458·19-s + 0.670·20-s − 1.27·22-s − 1.25·23-s + 4/5·25-s + 0.196·26-s − 1.67·29-s + 1.79·31-s + 0.176·32-s − 0.514·34-s − 1.15·37-s − 0.324·38-s + 0.474·40-s − 0.937·41-s − 0.609·43-s − 0.904·44-s − 0.884·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.43590489116254193591372166560, −6.42428355502892186171436272382, −6.16956002155740628421880966525, −5.22204405458644813686464103528, −5.01333912984247044922320835086, −3.93578360822709326417332802931, −3.00254246901057700707106974057, −2.21709797029952527285056762585, −1.75191805679101399207902224664, 0,
1.75191805679101399207902224664, 2.21709797029952527285056762585, 3.00254246901057700707106974057, 3.93578360822709326417332802931, 5.01333912984247044922320835086, 5.22204405458644813686464103528, 6.16956002155740628421880966525, 6.42428355502892186171436272382, 7.43590489116254193591372166560
