L(s) = 1 | − 3-s + 9-s + 2·11-s + 13-s + 2·17-s + 5·19-s + 6·23-s − 5·25-s − 27-s + 8·29-s + 3·31-s − 2·33-s + 9·37-s − 39-s − 2·41-s − 43-s − 8·47-s − 2·51-s − 6·53-s − 5·57-s + 6·59-s − 2·61-s + 5·67-s − 6·69-s + 4·71-s + 11·73-s + 5·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.485·17-s + 1.14·19-s + 1.25·23-s − 25-s − 0.192·27-s + 1.48·29-s + 0.538·31-s − 0.348·33-s + 1.47·37-s − 0.160·39-s − 0.312·41-s − 0.152·43-s − 1.16·47-s − 0.280·51-s − 0.824·53-s − 0.662·57-s + 0.781·59-s − 0.256·61-s + 0.610·67-s − 0.722·69-s + 0.474·71-s + 1.28·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.052656344\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.052656344\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.73481981725478303502885088863, −6.82624145186091263586709728995, −6.43821218151992390731994987113, −5.62435179683367215736880209608, −5.00772132876135053731225659468, −4.29907711338832695709318510416, −3.44202520810320032329145833957, −2.71801119342521337383665152130, −1.45342750093789786147458095293, −0.78584842393231782250292088133,
0.78584842393231782250292088133, 1.45342750093789786147458095293, 2.71801119342521337383665152130, 3.44202520810320032329145833957, 4.29907711338832695709318510416, 5.00772132876135053731225659468, 5.62435179683367215736880209608, 6.43821218151992390731994987113, 6.82624145186091263586709728995, 7.73481981725478303502885088863