| L(s) = 1 | − 3·9-s + 4·13-s + 4·17-s − 4·19-s − 8·23-s + 2·29-s + 8·31-s + 8·37-s − 6·41-s − 8·43-s + 8·47-s + 4·59-s + 6·61-s − 8·67-s + 12·71-s − 4·73-s − 4·79-s + 9·81-s + 10·89-s − 12·97-s + 18·101-s − 8·103-s − 8·107-s + 14·109-s + 16·113-s − 12·117-s + ⋯ |
| L(s) = 1 | − 9-s + 1.10·13-s + 0.970·17-s − 0.917·19-s − 1.66·23-s + 0.371·29-s + 1.43·31-s + 1.31·37-s − 0.937·41-s − 1.21·43-s + 1.16·47-s + 0.520·59-s + 0.768·61-s − 0.977·67-s + 1.42·71-s − 0.468·73-s − 0.450·79-s + 81-s + 1.05·89-s − 1.21·97-s + 1.79·101-s − 0.788·103-s − 0.773·107-s + 1.34·109-s + 1.50·113-s − 1.10·117-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
| \(L(1)\) |
\(\approx\) |
\(1.709262622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709262622\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.326979030386123550796107315511, −7.78002946262096741516646913645, −6.60205589186276157013670986996, −6.09860677460198488658964402707, −5.51882650234399560097838956504, −4.48696527340555482351004520803, −3.71591834868403946367514113426, −2.90833991340706952675630031412, −1.94961756315555168363313630573, −0.71313906915073102573118650858,
0.71313906915073102573118650858, 1.94961756315555168363313630573, 2.90833991340706952675630031412, 3.71591834868403946367514113426, 4.48696527340555482351004520803, 5.51882650234399560097838956504, 6.09860677460198488658964402707, 6.60205589186276157013670986996, 7.78002946262096741516646913645, 8.326979030386123550796107315511
