| L(s) = 1 | − 3·3-s − 7-s + 6·9-s − 11-s + 6·13-s − 3·17-s − 5·19-s + 3·21-s + 2·23-s − 9·27-s − 5·29-s + 5·31-s + 3·33-s + 37-s − 18·39-s − 2·41-s − 12·43-s + 2·47-s − 6·49-s + 9·51-s + 13·53-s + 15·57-s + 2·59-s + 61-s − 6·63-s − 16·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s − 0.301·11-s + 1.66·13-s − 0.727·17-s − 1.14·19-s + 0.654·21-s + 0.417·23-s − 1.73·27-s − 0.928·29-s + 0.898·31-s + 0.522·33-s + 0.164·37-s − 2.88·39-s − 0.312·41-s − 1.82·43-s + 0.291·47-s − 6/7·49-s + 1.26·51-s + 1.78·53-s + 1.98·57-s + 0.260·59-s + 0.128·61-s − 0.755·63-s − 1.95·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7426758511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7426758511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.043157029878232683903507916478, −8.346521097896039551534984477235, −7.19649494341533460696322097208, −6.36530496702041946022859029619, −6.14015315113938346252387880897, −5.16414604994050037819103052448, −4.39434795314299990970203380199, −3.48040092422512996734517779792, −1.86871175399850578232232099832, −0.61555083613053945518679201199,
0.61555083613053945518679201199, 1.86871175399850578232232099832, 3.48040092422512996734517779792, 4.39434795314299990970203380199, 5.16414604994050037819103052448, 6.14015315113938346252387880897, 6.36530496702041946022859029619, 7.19649494341533460696322097208, 8.346521097896039551534984477235, 9.043157029878232683903507916478
